INTEGRAL REPRESENTATION THEORY:applications to convexity, Banach spaces and

potential theory

J. Lukes, J. Maly, I. Netuka and J. Spurny

9th November 2009

Introduction

In many branches of mathematics, one encounters the question of how to reconstructa convex set from information on its vertices. This idea successfully emerged as theKrein–Milman theorem for compact convex subsets of locally convex spaces sinceany such set has plenty of extreme points. For any point of a compact convex set, areformulation of the Krein–Milman theorem provides a representing measure that isconcentrated in some sense on the set of extreme points. The goal of our book is topresent a more general approach to integral representationtheory based upon a notionof a function space and apply the obtained results to the theory of convex sets, Banachspaces and potential theory.

We point out that this approach is far from being new, but we hope that our ex-position may be profitable both for students interested in the basics of integral rep-resentation theory as well as for more advanced readers. Theformer group could beattracted by a self-contained presentation of the Choquet theory, the latter by a sub-stantial amount of results of fairly recent origin or appearing in a book form for thefirst time. We also try to incorporate more techniques from descriptive set theory intosubject, which further supports our belief that the book will be worth reading even forthose well acquainted with the monographs by E. M. Alfsen [5],R. R. Phelps [374],Z. Semadeni [414], L. Asimow and A. J. Ellis [24] or V. P. Fonf,J. Lindenstrauss andR. R. Phelps [179].

Let us continue by looking briefly at the contents of the book.After a prologueon the Korovkin theorem, we present basic facts on the extremal structure of finite-dimensional compact convex sets. Then we move on to infinite-dimensional spacesand prove the Krein–Milman theorem and several of its consequences. The secondpart of Chapter 2 studies the concept of measure convex and measure extremal sets.

Chapter 3 is devoted to cornerstones of the Choquet theory offunctions spacessuch as the Choquet order and its properties and integral representation theorems dueto G. Choquet and E. Bishop and K. de Leeuw. Even though the results are standard,the key limiting process is established by means of the Simons lemma, which allowsus to present later on several of its applications. The chapter is finished by a discussionon deeper properties of the Choquet ordering.

The next chapter studies basic properties of affine functions on compact convex setsand characterizations of functions satisfying the barycentric formula. A link betweenthe theory of function spaces and compact convex sets startsto emerge at the end ofthe chapter.

Chapter 5 is crucial for the subsequent application of descriptive set theory; it de-scribes a hierarchy of Borel sets and functions in topological spaces and proves their

Introduction 3

basic properties. The most important fact is that many descriptive properties are sta-ble with respect to perfect mappings, which allows us to transfer abstract Borel affinefunctions to the setting of compact convex sets.

Simplicial function spaces are studied in Chapter 6. We discuss several classes ofsimplicial function spaces, namely the Bauer and Markov simplicial function spacesand spaces with boundary of typeFσ. Among other results, the abstract Dirichletproblem for continuous and non-continuous functions is considered. Choquet sim-plices are presented at the end of the chapter.

Next we generalize the basic concepts for function cones since they are indispens-able in potential theory. We focus in particular on ordered compact convex sets.

Analogues of faces in a non-convex setting, so-called Choquet sets, are investigatedin Chapter 8. The main result is a characterization of simplicial spaces by means ofChoquet sets.

Suitably chosen families of closed extremal sets generate interesting boundarytopologies on the set of extreme points. Chapter 9 studies these topologies and func-tions continuous with respect to them. It turns out that maximal measures inducemeasures on sets of extreme points that are regular with respect to boundary topolo-gies. The last section is devoted to a study of a facial topology and facially continuousfunctions.

Chapter 10 collects several deeper results on function spaces and compact convexsets. Among others, study of Shilov and James boundaries, Lazar’s improvement ofthe Banach–Stone theorem, results on automatic boundedness of affine and convexfunctions, embedding ofℓ1 in Banach spaces, metrizability of compact convex setsand their open images and some topological properties of theset of extreme points.

The Lazar selection theorem and its consequences occupy thefirst part of Chap-ter 11. The second part is devoted to a presentation of Debs’ proof of Talagrand’stheorem on measurable selectors.

Chapter 12 is concerned with two methods of constructing newfunction spaces:products and inverse limits. We show that both operations preserve simpliciality anddescribe resulting boundaries. The inverse limits lead to an interesting descriptionof metrizable simplices as inverse limits of finite-dimensional simplices. The generalresults are illustrated by a construction of the Poulsen simplex and a couple of compactconvex sets due to Talagrand.

In Chapter 13, general results from Choquet’s theory are applied to potential the-ory and several of its basic notions are investigated from this perspective. Importantfunction cones and spaces appearing in potential theory arestudied in detail, in par-ticular, in connection to various solution methods for the Dirichlet problem. Thefunctional analysis approach makes it possible to provide an interesting interpreta-tion, for instance, of balayage and regular points in terms of representing measuresand the Choquet boundary of suitable spaces and cones. The exposition covers poten-tial theory for the Laplace equation and the heat equation aswell as a more general

4 Introduction

setting (harmonic spaces, fine potential theory etc.).The final Chapter 14 presents several applications of the integral representation

theorems, such as for doubly stochastic matrices, the Riesz–Herglotz theorem, theLyapunov theorem on the range of a vector measure, the Stone–Weierstrass theorem,positive-definite functions and invariant and ergodic measures.

Each chapter concludes with a series of exercises with sketches of proofs and withconcluding notes and comments where we try to give precise references and due cred-its for the results presented in the main body of the text, anddiscuss additional mater-ial which is related to the topics of the chapter in question,but was not included withcomplete proofs. Open problems are also mentioned.

Since the presented material originates in an amalgamationof functional analy-sis, measure theory, topology, descriptive set theory and potential theory, we collectthe needed notions and facts in the Appendix, sometimes evenwith proofs. We se-lected the following books for each subject as the key references: W. Rudin [403]and M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucıa, J. Pelant and V. Zizler[173] for functional analysis, D. H. Fremlin [182], [181] and [183] for measure the-ory, R. Engelking [169] and K. Kuratowski [285] for topology, A. S. Kechris [262]and C. A. Rogers and J. E. Jayne [394] for descriptive set theory, D. H. Armitage andS. J. Gardiner [21] for classical potential theory and J. Bliedtner and W. Hansen [66]for abstract potential theory.

Next we point out what is omitted from the book. First, we focus on integral repre-sentation theorems for compact sets, and thus the readers interested in theory of setswith the Radon–Nikodym property are referred to R. D. Bourgin [82], and those inter-ested in Choquet theory in sets of measures are referred to G.Winkler [473]. Second,although we consider several geometric aspects of simplicial spaces, they are not atthe center of our attention. They are thoroughly investigated in H. E. Lacey [290] andP. Harmand, D. Werner and W. Werner [216]. Further, we do not pursue applicationsof integral representation theory inC∗-algebras and thus we refer the interested readerto E. M. Alfsen and F. W. Schultz [10] and [9], M. Rørdam [395] andM. Rørdam andE. Størmer [396], B. Blackadar [59] and H. Lin [303] and the references therein. Andlast but not least, our applications to potential theory do not require the full strengthof abstract potential theory and thus we restrict ourselvesto a less general frameworkthan the one presented in J. Bliedtner and W. Hansen [66].

Except on a few explicitly stated occasions, we consider only real vector spaces andapart from Chapter 9 we deal only with Hausdorff topologies and Radon measures.We use the standard notation and terminology:

• N, Q, Z, R, C denote the usual sets of numbers,

• Rez and Imz denote the real and imaginary part of a complex numberz, respec-tively,

• cA is the characteristic function of a setA (sometimes we write 1 for the charac-teristic function of a space),

Introduction 5

• A△B is the symmetric difference of setsA andB,

• Ac is the complement of a setA,

• f ∧ g, f ∨ g denote the infimum and supremum of functionsf, g, respectively,(usually they are considered pointwise),

• f+, f−, |f | denote the positive and negative parts, and absolute value of a func-tion f , respectively,

• f |A is the restriction of a functionf to a setA,

• if F is a system of functions,Fb andF+ are the families of all bounded andpositive elements fromF , respectively,

• ω0 andω1 are the first infinite and first uncountable ordinals, respectively,

• A, IntA, ∂A are the closure, interior and boundary of a setA in a topologicalspace, respectively,

• dist(A, B) denotes the distance of sets in a metric space,

• diamA is the diameter of a setA in a metric space,

• U(x, r), B(x, r) andS(x, r) are the open ball, closed ball and sphere centeredatx with radiusr > 0, respectively,

• coA and spanA are the convex and linear hull of a setA in a vector space,respectively,coA is the closed convex hull of a setA in a topological vectorspace,

• kerT denotes the kernel of an operator between linear spaces,

• BE , UE andSE are the closed unit ball, open unit ball and sphere of a normedlinear spaceE, respectively,

• E/F is the quotient space of a locally convex space with respect to a closedsubspaceF ,

• E ⊕ F is the sum of locally convex spacesE andF ,

• E∗ is the dual space of a topological linear spaceE,

• (x, y) stands for the scalar product of vectorsx, y in a Hilbert space,

• c0 is the space of sequences converging to 0,

• C(X) is the space of real-valued continuous functions on a topological spaceX,

• Cb(X) is the space of bounded continuous functions on a topological spaceX,

• ℓp andLp(µ), p ∈ [1,∞], are the usual Lebesgue spaces (see Section A.3).

• Cn(U), Cn, C∞(U), C∞ stand for the functions onU with n-th derivative con-tinuous and for infinitely differentiable functions onU , respectively,

• −∫A

f(y) dy is the integral mean value off over a setA,

•

∫S(x,r) f(y) dS(y) is the surface integral off over the sphereS(x, r) ⊂ Rd,

6 Introduction

• ∇f is the gradient off .

A function f is positiveif f ≥ 0, it is strictly positiveif f > 0. Similarly we useincreasing, strictly increasingand so on. Ifµ is a measure, we often writeµ(f) forthe integral

∫f dµ.

In preparation of the present book, we have received many valuable suggestionsfrom many colleagues. In particular, we would like to express our thanks to P. Hajek,P. Holicky, M. Johanis, O. Kalenda, P. Kaplicky, M. Kraus, E. Murtinova, P. Simon,J. Tiser, L. Zajıcek and M. Zeleny for stimulating and fruitful discussions, and toE. Crooks for linguistic assistance. We are also indebted tothe publishers for theircare and cooperation.

The preparation of the manuscript was supported by the grant201/07/0388 of theGrant Agency of the Czech Republic and partly by the grant MSM21620839 of theCzech Ministry of Education.

Finally, our thanks go to Jana, Jarka, Hana and Hanka for encouragement and pa-tience during the preparation of this book.

Prague, Summer, 2009 Jaroslav Lukes, Jan Maly, Ivan Netuka and Jirı Spurny

Contents

Introduction 2

1 Prologue 11.1 The Korovkin theorem . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Compact convex sets 42.1 Geometry of convex sets . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.A Finite-dimensional case . . . . . . . . . . . . . . . . . . . . 52.1.B The Krein–Milman theorem . . . . . . . . . . . . . . . . . . 92.1.C Exposed points . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Interlude: On the spaceM(K) . . . . . . . . . . . . . . . . . . . . . 222.3 Structures in convex sets . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.A Extremal sets and faces . . . . . . . . . . . . . . . . . . . . . 262.3.B Measure convex sets . . . . . . . . . . . . . . . . . . . . . . 302.3.C Measure extremal sets . . . . . . . . . . . . . . . . . . . . . 36

2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Choquet theory of function spaces 523.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 More about Korovkin theorems . . . . . . . . . . . . . . . . . . . . . 643.3 On theH-barycenter mapping . . . . . . . . . . . . . . . . . . . . . 663.4 The Choquet representation theorem . . . . . . . . . . . . . . . . .. 673.5 In-between theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 703.6 Maximal measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.7 Boundaries and the Simons lemma . . . . . . . . . . . . . . . . . . . 783.8 The Bishop–de Leeuw theorem . . . . . . . . . . . . . . . . . . . . . 813.9 Minimum principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.10 Orderings and dilations . . . . . . . . . . . . . . . . . . . . . . . . . 863.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.12 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Affine functions on compact convex sets 1084.1 Affine functions and the barycentric formula . . . . . . . . . .. . . . 1084.2 Barycentric theorem and strongly affine functions . . . . .. . . . . . 114

8 Contents

4.3 State space and representation of affine functions . . . . .. . . . . . 1214.4 Affine Baire-one functions on dual unit balls . . . . . . . . . .. . . . 1284.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5 Perfect classes of functions and representation of affine functions 1365.1 Generation of sets and functions . . . . . . . . . . . . . . . . . . . .1375.2 Baire and Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.3 Baire and Borel mappings . . . . . . . . . . . . . . . . . . . . . . . 1475.4 Perfect classes of functions . . . . . . . . . . . . . . . . . . . . . . .1505.5 Affinely perfect classes of functions . . . . . . . . . . . . . . . .. . 1515.6 Representation ofH-affine functions . . . . . . . . . . . . . . . . . . 1555.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.8 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6 Simplicial function spaces 1696.1 Basic properties of simplicial spaces . . . . . . . . . . . . . . .. . . 1706.2 Characterizations of simplicial spaces . . . . . . . . . . . . .. . . . 1776.3 Simplicial spaces asL1-preduals . . . . . . . . . . . . . . . . . . . . 1796.4 The weak Dirichlet problem andAc(H)-exposed points . . . . . . . . 1816.5 The Dirichlet problem for a single function . . . . . . . . . . .. . . 1836.6 Special classes of simplicial spaces . . . . . . . . . . . . . . . .. . . 186

6.6.A Bauer simplicial spaces . . . . . . . . . . . . . . . . . . . . . 1866.6.B Markov simplicial spaces . . . . . . . . . . . . . . . . . . . . 1896.6.C Simplicial spaces with Lindelof boundaries . . . . . . . . . . 1916.6.D Simplicial spaces with boundaries of typeFσ . . . . . . . . . 193

6.7 The Daugavet property of simplicial spaces . . . . . . . . . . .. . . 1976.8 Choquet simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.8.A Simplicial function spaces and the classical definition of Cho-quet simplices . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.8.B Prime function spaces and prime compact convex sets . .. . 2016.8.C Characterization of Bauer simplices by faces . . . . . . .. . 2036.8.D Fakhoury’s theorem . . . . . . . . . . . . . . . . . . . . . . 204

6.9 Restriction of function spaces . . . . . . . . . . . . . . . . . . . . .. 2056.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2066.11 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7 Choquet theory of function cones 2177.1 Function cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2 Maximal measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.3 Representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2257.4 Simplicial cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Contents 9

7.5 Ordered compact convex sets and simplicial measures . . .. . . . . . 2337.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2417.7 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8 Choquet-like sets 2458.1 Split and parallel faces . . . . . . . . . . . . . . . . . . . . . . . . . 2458.2 H-extremal andH-convex sets . . . . . . . . . . . . . . . . . . . . . 2478.3 Choquet sets,M -sets andP -sets . . . . . . . . . . . . . . . . . . . . 2518.4 H-exposed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2588.5 Weak topology on boundary measures . . . . . . . . . . . . . . . . . 2608.6 Characterizations of simpliciality by Choquet sets . . .. . . . . . . . 2638.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.8 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 274

9 Topologies on boundaries 2759.1 Topologies generated by extremal sets . . . . . . . . . . . . . . .. . 2759.2 Induced measures on Choquet boundaries . . . . . . . . . . . . . .. 2809.3 Functions continuous inσext andσmax topologies . . . . . . . . . . . 2859.4 Strongly universally measurable functions . . . . . . . . . .. . . . . 2899.5 Facial topology generated byM -sets . . . . . . . . . . . . . . . . . . 2989.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3049.7 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 309

10 Deeper results on function spaces and compact convex sets 31110.1 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10.1.A Shilov boundary . . . . . . . . . . . . . . . . . . . . . . . . 31210.1.B Boundaries in Banach spaces . . . . . . . . . . . . . . . . . . 315

10.2 Isometries of spaces of affine continuous functions . . .. . . . . . . 32110.3 Baire measurability and boundedness of affine functions . . . . . . . 324

10.3.A The Cantor set and its properties . . . . . . . . . . . . . . . . 32410.3.B Automatic boundedness of affine and convex functions. . . . 329

10.4 Embedding ofℓ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33610.5 Metrizability of compact convex sets . . . . . . . . . . . . . . . .. . 34010.6 Continuous affine images . . . . . . . . . . . . . . . . . . . . . . . . 35210.7 Several topological results on Choquet boundaries . . .. . . . . . . . 359

10.7.A The Choquet boundary as a Baire space . . . . . . . . . . . . 35910.7.B Polish spaces as Choquet boundaries . . . . . . . . . . . . . .36010.7.C K-countably determined boundaries . . . . . . . . . . . . . . 365

10.8 Convex Baire-one functions . . . . . . . . . . . . . . . . . . . . . . 36610.9 Function spaces with continuous envelopes . . . . . . . . . .. . . . 371

10.9.A Stable compact convex sets . . . . . . . . . . . . . . . . . . . 37110.9.B CE-function spaces . . . . . . . . . . . . . . . . . . . . . . . 377

10 Contents

10.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37910.11Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 385

11 Continuous and measurable selectors 39011.1 The Lazar selection theorem . . . . . . . . . . . . . . . . . . . . . . 39011.2 Applications of the Lazar selection theorem . . . . . . . . .. . . . . 39511.3 The weak Dirichlet problem for Baire functions . . . . . . .. . . . . 39911.4 Pointwise approximation of maximal measures . . . . . . . .. . . . 40111.5 Measurable selectors . . . . . . . . . . . . . . . . . . . . . . . . . . 403

11.5.A Multivalued mappings . . . . . . . . . . . . . . . . . . . . . 40311.5.B Selection theorem . . . . . . . . . . . . . . . . . . . . . . . . 40711.5.C Applications of the selection theorem . . . . . . . . . . . .. 410

11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41311.7 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 418

12 Constructions of function spaces 42012.1 Products of function spaces . . . . . . . . . . . . . . . . . . . . . . .421

12.1.A Definitions and basic properties . . . . . . . . . . . . . . . . 42112.1.B Maximal measures and extremal sets . . . . . . . . . . . . . 42512.1.C Partitions of unity and approximation in products offunction

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42912.1.D Products of simplicial spaces . . . . . . . . . . . . . . . . . . 437

12.2 Inverse limits of function spaces . . . . . . . . . . . . . . . . . .. . 44112.2.A Admissible mappings . . . . . . . . . . . . . . . . . . . . . . 44112.2.B Construction of inverse limits . . . . . . . . . . . . . . . . . 44312.2.C Inverse limits of simplicial function spaces . . . . . .. . . . 44612.2.D Structure of simplices . . . . . . . . . . . . . . . . . . . . . 448

12.3 Several examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45612.3.A The Poulsen simplex . . . . . . . . . . . . . . . . . . . . . . 45612.3.B A big simplicial space . . . . . . . . . . . . . . . . . . . . . 46612.3.C Functions of affine classes and Talagrand’s example .. . . . 471

12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47712.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 487

13 Function spaces in potential theory and the Dirichlet problem 49113.1 Balayage and the Dirichlet problem . . . . . . . . . . . . . . . . .. 493

13.1.A Essential solution of the generalized Dirichlet problem . . . . 49613.2 Boundary behavior of solutions . . . . . . . . . . . . . . . . . . . .. 497

13.2.A Regular points for the Laplace equation . . . . . . . . . . .. 49913.2.B Regular points for the heat equation . . . . . . . . . . . . . .504

13.3 Function spaces and cones in potential theory . . . . . . . .. . . . . 50513.3.A Function spaces and cones: Laplace equation . . . . . . .. . 507

Contents 11

13.3.B Function spaces and cones in parabolic potential theory andharmonic spaces . . . . . . . . . . . . . . . . . . . . . . . . 511

13.3.C Continuity properties ofH(U)-concave functions . . . . . . . 51513.3.D Separation by functions fromH(U) . . . . . . . . . . . . . . 517

13.4 Dirichlet problem: solution methods . . . . . . . . . . . . . . .. . . 51913.4.A PWB solution of the Dirichlet problem . . . . . . . . . . . . 51913.4.B Cornea’s approach to the Dirichlet problem . . . . . . . .. . 52413.4.C The Wiener solution . . . . . . . . . . . . . . . . . . . . . . 53113.4.D Fine Wiener solution . . . . . . . . . . . . . . . . . . . . . . 53313.4.E PDE solutions in Sobolev spaces . . . . . . . . . . . . . . . . 535

13.5 Generalized Dirichlet problem and uniqueness questions . . . . . . . 53813.5.A Lattice approach . . . . . . . . . . . . . . . . . . . . . . . . 53913.5.B Uniqueness for the Laplace equation . . . . . . . . . . . . . .54113.5.C Keldysh theorems in parabolic and axiomatic potential theories 543

13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54713.7 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 557

14 Applications 56514.1 Representation of convex functions . . . . . . . . . . . . . . . .. . . 56614.2 Representation of concave functions . . . . . . . . . . . . . . .. . . 56914.3 Doubly stochastic matrices . . . . . . . . . . . . . . . . . . . . . . .57414.4 The Riesz–Herglotz theorem . . . . . . . . . . . . . . . . . . . . . . 57514.5 Typically real holomorphic functions . . . . . . . . . . . . . .. . . . 57714.6 Holomorphic functions with positive real part . . . . . . .. . . . . . 58214.7 Completely monotonic functions . . . . . . . . . . . . . . . . . . .. 58814.8 Positive definite functions on discrete groups . . . . . . .. . . . . . 59114.9 Range of vector measures . . . . . . . . . . . . . . . . . . . . . . . . 59514.10The Stone–Weierstrass approximation theorem . . . . . .. . . . . . 59714.11Invariant and ergodic measures . . . . . . . . . . . . . . . . . . .. . 59914.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60514.13Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 607

A Appendix 610A.1 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

A.1.A Locally convex spaces . . . . . . . . . . . . . . . . . . . . . 610A.1.B Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 611A.1.C Ordered Banach spaces and lattices . . . . . . . . . . . . . . 612

A.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617A.2.A Compact spaces andCech–Stone compactification . . . . . . 618A.2.B Baire and Borel sets . . . . . . . . . . . . . . . . . . . . . . 621A.2.C Semicontinuous functions . . . . . . . . . . . . . . . . . . . 623A.2.D Baire spaces and sets with the Baire property . . . . . . . .. 625

12 Contents

A.3 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626A.3.A Measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . 626A.3.B Radon measures on locally compactσ-compact spaces . . . . 628A.3.C Images, products and inverse limits of Radon measures. . . . 635A.3.D Kernels and disintegration of measures . . . . . . . . . . . .638

A.4 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 639A.5 Resolvable sets and Baire-one functions . . . . . . . . . . . . .. . . 642A.6 The Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . 647

A.6.A Weak solutions of the Laplace equation . . . . . . . . . . . . 649A.7 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651A.8 Axiomatic potential theory . . . . . . . . . . . . . . . . . . . . . . . 654

A.8.A Bauer’s axiomatic theory . . . . . . . . . . . . . . . . . . . . 655A.8.B Hyperharmonic and superharmonic functions . . . . . . . .. 656A.8.C Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658A.8.D Superharmonic functions and Green potentials . . . . . .. . 659A.8.E Superharmonic functions and potentials for the heat equation . 662A.8.F Balayage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663A.8.G Thinness, base and fine topology . . . . . . . . . . . . . . . . 665A.8.H Polar and semipolar sets . . . . . . . . . . . . . . . . . . . . 667

Bibliography 671

List of symbols 697

Bibliography

[1] E. M. Alfsen, On the geometry of Choquet simplexes,Math. Scand.15 (1964), 97–110.

[2] , Boundary values for homomorphisms of compact convex sets,Math. Scand.19 (1966), 113–121.

[3] , Facial structure of compact convex sets,Proc. London Math. Soc. (3)18(1968), 385–404.

[4] , On the Dirichlet problem of the Choquet boundary,Acta Math.120 (1968),149–159.

[5] , Compact convex sets and boundary integrals, Springer-Verlag, New York,1971, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57.

[6] E. M. Alfsen and T. B. Andersen, Split faces of compact convex sets,Arhus University(1968/69).

[7] , Split faces of compact convex sets,Proc. London Math. Soc. (3)21 (1970),415–442.

[8] E. M. Alfsen and B. Hirsberg, On dominated extensions in linear subspaces ofCC (X),Pacific J. Math.36 (1971), 567–584.

[9] E. M. Alfsen and F. W. Shultz,State spaces of operator algebras, Mathematics: Theory& Applications, Birkhauser Boston Inc., Boston, MA, 2001, Basic theory, orientations,andC∗-products.

[10] , Geometry of state spaces of operator algebras, Mathematics: Theory & Ap-plications, Birkhauser Boston Inc., Boston, MA, 2003.

[11] E. M. Alfsen and C. F. Skau, Simplicial decomposition of boundary measures on con-vex compact sets.,Math. Scand.26 (1970), 62–72.

[12] S. Alpay, A localization of the equal support property,J. Pure Appl. Sci.11 (1978),233–235 (1980).

[13] F. Altomare and M. Campiti,Korovkin-type approximation theory and its applications,de Gruyter Studies in Mathematics 17, Walter de Gruyter & Co., Berlin, 1994, Appen-dix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff.

[14] D. Amir, On isomorphisms of continuous function spaces,Israel J. Math.3 (1965),205–210.

[15] P. R. Andenaes, Extreme boundaries and continuous affine functions,Math. Scand.40(1977), 197–208.

[16] T. B. Andersen and H. R. Atkinson, On Banach algebra-valued facially continuousfunctions,J. London Math. Soc. (2)9 (1974/75), 381–384.

672 Bibliography

[17] R. Arens, Extension of functions on fully normal spaces,Pacific J. Math.2 (1952),11–22.

[18] R. F. Arens and J. L. Kelley, Characterization of the space of continuous functions overa compact Hausdorff space,Trans. Amer. Math. Soc.62 (1947), 499–508.

[19] S. A. Argyros, G. Godefroy and H. P. Rosenthal,Descriptive set theory and Banachspaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amster-dam, 2003, pp. 1007–1069.

[20] D. H. Armitage,The Riesz-Herglotz representation for positive harmonic functionsvia Choquet’s theorem, Potential theory—ICPT 94 (Kouty, 1994), de Gruyter, Berlin,1996, pp. 229–232.

[21] D. H. Armitage and S. J. Gardiner,Classical potential theory, Springer Monographs inMathematics, Springer-Verlag London Ltd., London, 2001.

[22] M. G. Arsove, The Wiener-Dirichlet problem and the theorem of Evans,Math. Z.103(1968), 184–194.

[23] E. Artin, A proof of the Krein–Milman Theorem, Collected papers, Springer-Verlag,New York, 1982, Edited by Serge Lang and John T. Tate, Reprint of the 1965 original.

[24] L. Asimow and A. J. Ellis,Convexity theory and its applications in functional analysis,London Mathematical Society Monographs 16, Academic Press Inc. [Harcourt BraceJovanovich Publishers], London, 1980.

[25] R. E. Atalla, Markov operators, peak points, and Choquet points,Proc. Amer. Math.Soc.41 (1973), 103–109.

[26] I. Babuska and R. Vyborny, Regulare und stabile Randpunkte fur das Problem derWarmeleitungsgleichung,Ann. Polon. Math.12 (1962), 91–104.

[27] M. Bacak, Point simpliciality in Choquet representation theory,Illinois J. Math., toappear.

[28] M. Bacak and J. Spurny, Complementability of spaces of affine continuous functionson simplices,Bull. Belg. Math. Soc. Simon Stevin15 (2008), 465–472.

[29] R. Baire, Sur les fonctions de variables relees,Ann. di Mat. Pura ed Appl.3 (1899),1–123.

[30] M. V. Balashov, An analogue of the Kreın-Milman theorem for a strongly convex hullin a Hilbert space,Mat. Zametki71 (2002), 37–42.

[31] A. Barvinok, A course in convexity, Graduate Studies in Mathematics 54, AmericanMathematical Society, Providence, RI, 2002.

[32] C. J. K. Batty, Maximal measures on tensor products of compact convex sets, Quart. J.Math. Oxford Ser. (2)33 (1982), 1–10.

[33] , A characterisation of simplexes by an extension property,Quart. J. Math.Oxford Ser. (2)34 (1983), 391–397.

[34] , Some properties of maximal measures on compact convex sets,Math. Proc.Cambridge Philos. Soc.94 (1983), 297–305.

Bibliography 673

[35] , Topologies and continuous functions on extreme points and pure states,Math.Proc. Cambridge Philos. Soc.98 (1985), 501–511.

[36] H. Bauer, Minimalstellen von Funktionen und Extremalpunkte,Arch. Math.9 (1958),389–393.

[37] , Un probleme de Dirichlet pour la frontiere deSilov d’un space compact,C.R. Acad. Sci. Paris247 (1958), 843–846.

[38] , Silovscher Rand und Dirichletsches Problem,Ann. Inst. Fourier Grenoble11(1961), 89–136, XIV.

[39] , Axiomatische Behandlung des Dirichletschen Problems fur elliptische undparabolische Differentialgleichungen,Math. Ann.146 (1962), 1–59.

[40] , Harmonische Raume und ihre Potentialtheorie, Ausarbeitung einer im Som-mersemester 1965 an der Universitat Hamburg gehaltenen Vorlesung. Lecture Notes inMathematics, No. 22, Springer-Verlag, Berlin, 1966.

[41] , Aspects of modern potential theory, in:Proceedings of the InternationalCongress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 41–51, Canad. Math.Congress, Montreal, Que., 1975.

[42] , Approximation and abstract boundaries,Amer. Math. Monthly85 (1978),632–647.

[43] , Harmonic spaces—a survey,Confer. Sem. Mat. Univ. Bari(1984), 34.

[44] , Simplicial function spaces and simplexes,Exposition. Math.3 (1985), 165–168.

[45] , Measure and integration theory, de Gruyter Studies in Mathematics 26, Walterde Gruyter & Co., Berlin, 2001, Translated from the German by Robert B. Burckel.

[46] H. S. Bear, The Silov boundary for a linear space of continuous functions,Amer. Math.Monthly68 (1961), 483–485.

[47] R. Becker,Convex cones in analysis, Travaux en Cours [Works in Progress] 67, Her-mannEditeurs des Sciences et des Arts, Paris, 2006, With a postface by G. Choquet,Translation of the 1999 French version.

[48] J. B. Bednar, On the Dirichlet problem for functions on the extreme boundary of acompact convex set,Math. Scand.27 (1970), 141–144 (1971).

[49] E. Behrends and G. Wittstock, Tensorprodukte kompakter konvexer Mengen,Invent.Math.10 (1970), 251–266.

[50] , Tensorprodukte und Simplexe,Invent. Math.11 (1970), 188–198.

[51] D. Bensimon, The topological space of all extreme points of a compact convex set,Rend. Circ. Mat. Palermo (2)37 (1988), 177–200.

[52] Y. Benyamini and J. Lindenstrauss, A predual ofl1 which is not isomorphic to aC(K)space, in:Proceedings of the International Symposium on Partial Differential Equa-tions and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 13, pp. 246–254(1973), 1972.

674 Bibliography

[53] S. K. Berberian, Compact convex sets in inner product spaces,Amer. Math. Monthly74(1967), 702–705.

[54] C. Berg, J. P. Reus Christensen and P. Ressel,Harmonic analysis on semigroups, Grad-uate Texts in Mathematics 100, Springer-Verlag, New York, 1984, Theory of positivedefinite and related functions.

[55] S. Bernstein, Sur les fonctions absolument monotones,Acta Math.52 (1929), 1–66.

[56] C. Bessaga and T. Dobrowolski, Affine and homeomorphic embeddings intol2, Proc.Amer. Math. Soc.125 (1997), 259–268.

[57] G. Birkhoff, Three observations on linear algebra,Univ. Nac. Tucuman. Revista A.5(1946), 147–151.

[58] E. Bishop and K. de Leeuw, The representations of linear functionals by measures onsets of extreme points,Ann. Inst. Fourier. Grenoble9 (1959), 305–331.

[59] B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences 122,Springer-Verlag, Berlin, 2006, Theory ofC∗-algebras and von Neumann algebras, Op-erator Algebras and Non-commutative Geometry, III.

[60] W. Blaschke and G. Pick, Distanzschatzungen im Funktionenraum II,Math. Ann.77(1916), 277–300.

[61] J. Bliedtner,Approximation by harmonic functions, Potential theory—ICPT 94 (Kouty,1994), de Gruyter, Berlin, 1996, pp. 297–302.

[62] J. Bliedtner and W. Hansen, Simplicial cones in potential theory,Invent. Math.29(1975), 83–110.

[63] , Cones of hyperharmonic functions,Math. Z.151 (1976), 71–87.

[64] , Simplicial cones in potential theory. II. Approximation theorems,Invent.Math.46 (1978), 255–275.

[65] , The weak Dirichlet problem,J. Reine Angew. Math.348 (1984), 34–39.

[66] , Potential theory, Universitext, Springer-Verlag, Berlin, 1986, An analytic andprobabilistic approach to balayage.

[67] N. Boboc and Gh. Bucur, Sur le prolongement des fonctions affines,Math. Scand.26(1970), 42–50.

[68] , Cones convexes de fonctions continues sur un espace compact. Topologiessur la frontiere de Choquet,Rev. Roumaine Math. Pures Appl.17 (1972), 1307–1316.(1 plate), Collection of articles dedicated to G. Calugareanu on his seventieth birthday.

[69] , Simplicial measures on a compact space,Rev. Roumaine Math. Pures Appl.17 (1972), 1155–1163.

[70] , Conuri convexe de functii continue pe spatii compacte, Editura AcademieiRepublicii Socialiste Romania, Bucharest, 1976, With an English summary.

[71] N. Boboc, Gh. Bucur and A. Cornea,Order and convexity in potential theory:H-cones, Lecture Notes in Mathematics 853, Springer, Berlin, 1981, In collaboration withHerbert Hollein.

Bibliography 675

[72] N. Boboc, C. Constantinescu and A. Cornea, On the Dirichlet problem in the axiomatictheory of harmonic functions,Nagoya Math. J.23 (1963), 73–96.

[73] N. Boboc and A. Cornea, Convex cones of lower semicontinuous functions on compactspaces,Rev. Roumaine Math. Pures Appl.12 (1967), 471–525.

[74] H. Bohman, On approximation of continuous and of analytic functions,Ark. Mat. 2(1952), 43–56.

[75] F. F. Bonsall, On the representation of points of a convex set,J. London Math. Soc.38(1963), 332–334.

[76] V. Borovikov, On the intersection of a sequence of simplexes,Uspehi Matem. Nauk(N.S.)7 (1952), 179–180.

[77] K. Borsuk,Uber Isomorphie der Funktionalraume,Bull. Acad. Polonaise(1933), 1–10.

[78] G. Bouligand, Sur le probleme de Dirichlet,Ann. Soc. Polon. Math.4 (1926), 59–112.

[79] N. Bourbaki,Integration. I. Chapters 1–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2004, Translated from the 1959, 1965 and 1967 French originals bySterling K. Berberian.

[80] J. Bourgain, D. H. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions,Amer. J. Math.100 (1978), 845–886.

[81] J. Bourgain and M. Talagrand, Compacite extremale,Proc. Amer. Math. Soc.80 (1980),68–70.

[82] R. D. Bourgin,Geometric aspects of convex sets with the Radon-Nikodym property,Lecture Notes in Mathematics 993, Springer-Verlag, Berlin, 1983.

[83] M. Brelot, Familles de Perron et probleme de Dirichlet,Acta Litt. Sci. Szeged9 (1939),133–153.

[84] , Minorantes sous-harmoniques, extremales et capacites,J. Math. Pures Appl.(9) 24 (1945), 1–32.

[85] , Remarque sur le prolongement fonctionnel lineaire et le probleme de Dirich-let, Acta Sci. Math. Szeged12 (1950), 150–152.

[86] , Sur un theoreme de prolongement fonctionnel de Keldych concernant leprobleme de Dirichlet,J. Analyse Math.8 (1960/1961), 273–288.

[87] , Elements de la theorie classique du potentiel, 3e edition. Les cours de Sor-bonne. 3e cycle, Centre de Documentation Universitaire, Paris, 1965.

[88] , Historical introduction, Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969),Edizioni Cremonese, Rome, 1970, pp. 1–21.

[89] , Les etapes et les aspects multiples de la theorie du potentiel,EnseignementMath. (2)18 (1972), 1–36.

[90] , Le balayage de Poincare et l’epine de Lebesgue, in:Proceedings of the 110thnational congress of learned societies (Montpellier, 1985), pp. 141–151, Com. Trav.Hist. Sci., Paris, 1985.

676 Bibliography

[91] E. Briem, Facial topologies for subspaces ofC(X), Math. Ann.208 (1974), 9–13.

[92] , Extreme orthogonal boundary measures forA(K) and decompositions forcompact convex sets, Spaces of analytic functions (Sem. Functional Anal. and FunctionTheory, Kristiansand, 1975), Springer, Berlin, 1976, pp. 8–16. Lecture Notes in Math.,Vol. 512.

[93] , Enlarging a subspace ofC(X) without changing the Choquet boundary,Math.Scand.44 (1979), 218–224.

[94] , A characterization of simplexes by parallel faces,Bull. London Math. Soc.12(1980), 55–59.

[95] R. B. Burckel,An introduction to classical complex analysis. Vol. 1, Pure and AppliedMathematics 82, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], NewYork, 1979.

[96] M. Capon, Sur les fonctions qui verifient le calcul barycentrique,Proc. London Math.Soc. (3)32 (1976), 163–180.

[97] P. Cartier, J. M. G. Fell and P.-A. Meyer, Comparaison des mesures portees par unensemble convexe compact,Bull. Soc. Math. France92 (1964), 435–445.

[98] B. Cascales and G. Godefroy, Angelicity and the boundary problem,Mathematika45(1998), 105–112.

[99] B. Cascales, G. Manjabacas and G. Vera, A Krein-Smulian type result in Banachspaces,Quart. J. Math. Oxford Ser. (2)48 (1997), 161–167.

[100] B. Cascales and R. Shvydkoy, On the Krein-Smulian theorem for weaker topologies,Illinois J. Math.47 (2003), 957–976.

[101] B. Cascales and G. Vera, Topologies weaker than the weak topology of a Banachspace,J. Math. Anal. Appl.182 (1994), 41–68.

[102] C. Castaing and M. Valadier,Convex analysis and measurable multifunctions, LectureNotes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977.

[103] M. M. Choban, On some problems of descriptive set theory in topological spaces,Us-pekhi Mat. Nauk60 (2005), 123–144.

[104] Corson H. Choquet, G. and V. Klee, Exposed points of convex sets,Pacific J. Math.17(1966), 33–43.

[105] G. Choquet, Theory of capacities,Ann. Inst. Fourier, Grenoble5 (1953–1954), 131–295 (1955).

[106] , Le theoreme de representation integrale dans les ensembles convexes com-pacts,Ann. Inst. Fourier Grenoble10 (1960), 333–344.

[107] , Remarquea propos de la demonstration de l’unicite de P.-A. Meyer,SeminaireBrelot–Choquet–Deny (Theorie de Potentiel)6 (1961/62), Expose No. 8.

[108] , Deux exemples classiques de representation integrale,Enseignement Math.(2) 15 (1969), 63–75.

Bibliography 677

[109] , Lectures on analysis. Vol. I–III: Infinite dimensional measures and problemsolutions, Edited by J. Marsden, T. Lance and S. Gelbart, W. A. Benjamin, Inc., NewYork-Amsterdam, 1969.

[110] , La naissance de la theorie des capacites: reflexion sur une experience person-nelle,C. R. Acad. Sci. Ser. Gen. Vie Sci.3 (1986), 385–397.

[111] , Une demonstration du theoreme de Bochner-Weil par discretisation dugroupe,Results Math.9 (1986), 1–9.

[112] , Existence et unicite des representations integrales au moyen des pointsextremaux dans les cones convexes, Seminaire Bourbaki, Vol. 4, Soc. Math. France,Paris, 1995, pp. Exp. No. 139, 33–47.

[113] G. Choquet and P.-A. Meyer, Existence et unicite des representations integrales dansles convexes compacts quelconques,Ann. Inst. Fourier (Grenoble)13 (1963), 139–154.

[114] J. P. R. Christensen, Borel structures and a topological zero-one law,Math. Scand.29(1971), 245–255 (1972).

[115] , Compact convex sets and compact Choquet simplexes,Invent. Math.19(1973), 1–4.

[116] , Topology and Borel structure, North-Holland Publishing Co., Amsterdam,1974, Descriptive topology and set theory with applications to functional analysis andmeasure theory, North-Holland Mathematics Studies, Vol. 10. (Notas de Matematica,No. 51).

[117] C. H. Chu, Anti-lattices and prime sets,Math. Scand.31 (1972), 151–165.

[118] , A note on faces of compact convex sets,J. London Math. Soc. (2)5 (1972),556–560.

[119] C. H. Chu and H. B. Cohen, Isomorphisms of spaces of continuous affine functions,Pacific J. Math.155 (1992), 71–85.

[120] A. Clausing and G. Magerl, Generalized Dirichlet problems and continuous selectionsof representing measures,Math. Ann.216 (1975), 71–78.

[121] A. Clausing and S. Papadopoulou, Stable convex sets and extremal operators,Math.Ann.231 (1977/78), 193–203.

[122] H. B. Cohen, A bound-two isomorphism betweenC(X) Banach spaces,Proc. Amer.Math. Soc.50 (1975), 215–217.

[123] C. Constantinescu and A. Cornea,Potential theory on harmonic spaces, Springer-Verlag, New York, 1972, With a preface by H. Bauer, Die Grundlehren der mathe-matischen Wissenschaften, Band 158.

[124] A. Cornea, Resolution du probleme de Dirichlet et comportement des solutionsa lafrontiere a l’aide des fonctions de controle, C. R. Acad. Sci. Paris Ser. I Math. 320(1995), 159–164.

[125] , Applications of controlled convergence in analysis, Analysis and topology,World Sci. Publ., River Edge, NJ, 1998, pp. 257–275.

678 Bibliography

[126] H. H. Corson, A compact convex set inE3 whose exposed points are of the first cate-gory,Proc. Amer. Math. Soc.16 (1965), 1015–1021.

[127] , Metrizability of compact convex sets,Trans. Amer. Math. Soc.151 (1970),589–596.

[128] A. Curnock, J. Howroyd and Ngai-Ching Wong,The unique decomposition propertyand the Banach-Stone theorem, Function spaces (Edwardsville, IL, 2002), Contemp.Math. 328, Amer. Math. Soc., Providence, RI, 2003, pp. 151–156.

[129] P. C. Curtis, Jr.,On a theorem of Keldysh and Wiener, Abstract Spaces and Approxi-mation (Proc. Conf., Oberwolfach, 1968), Birkhauser, Basel, 1969, pp. 351–356.

[130] L. Danzer, B. Grunbaum and V. Klee,Helly’s theorem and its relatives, Proc. Sympos.Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 101–180.

[131] I. K. Daugavet, A property of completely continuous operators in the spaceC, UspehiMat. Nauk18 (1963), 157–158.

[132] E. B. Davies and G. F. Vincent-Smith, Tensor products, infinite products, and projectivelimits of Choquet simplexes,Math. Scand.22 (1968), 145–164 (1969).

[133] L. de Branges, The Stone-Weierstrass theorem,Proc. Amer. Math. Soc.10 (1959),822–824.

[134] A. de la Pradelle,A propos du memoire de G. F. Vincent-Smith sur l’approximationdes fonctions harmoniques,Ann. Inst. Fourier (Grenoble)19 (1969), 355–370 (1970).

[135] , Approximation des fonctions harmoniquesa l’aide d’un theoreme de G. F.Vincent-Smith, Seminaire de Theorie du Potentiel, dirige par M. Brelot, G. Choquet etJ. Deny (1969/70), Exp. 3, Secretariat Mathematique, Paris, 1971, p. 16.

[136] M. De Wilde, Pointwise compactness in spaces of functions and R. C. James theorem,Math. Ann.208 (1974), 33–47.

[137] G. Debs,Selections maximales d’une multi–application, Unpublished manuscript.

[138] , Applications affines ouvertes et convexes compacts stables,Bull. Sci. Math.(2) 102 (1978), 401–414.

[139] , Selections maximales d’une multi–application,Le seminaire d’Initiation al’Analyse29 (1979).

[140] , Some general methods for constructing stable convex sets,Math. Ann.241(1979), 97–105.

[141] , Quelques proprietes des espacesα-favorables et applications aux convexescompacts,Ann. Inst. Fourier (Grenoble)30 (1980), vii, 29–43.

[142] C. Dellacherie,Un complement au theoreme de Weierstrass-Stone, Seminaire de Prob-abilites (Univ. Strasbourg, Strasbourg, 1966/67), Vol. I, Springer, Berlin, 1967, pp. 52–53.

[143] R. Deville and C. Finet, An extension of Simons’ inequality and applications,Rev. Mat.Complut.14 (2001), 95–104.

Bibliography 679

[144] J. L. Doob, Generalized sweeping-out and probability,J. Functional Analysis2 (1968),207–225.

[145] , Classical potential theory and its probabilistic counterpart, Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]262, Springer-Verlag, New York, 1984.

[146] P. Dostal, J. Lukes and J. Spurny, Measure convex and measure extremal sets,Canad.Math. Bull.49 (2006), 536–548.

[147] R. G. Douglas, On extremal measures and subspace density,Michigan Math. J.11(1964), 243–246.

[148] T. Downarowicz, The Choquet simplex of invariant measures for minimal flows,IsraelJ. Math.74 (1991), 241–256.

[149] J. Dugundji, An extension of Tietze’s theorem,Pacific J. Math.1 (1951), 353–367.

[150] N. Dunford and J. T. Schwartz,Linear operators. Part I, Wiley Classics Library, JohnWiley & Sons Inc., New York, 1988, General theory, With the assistance of William G.Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publi-cation.

[151] M. Edelstein and J. E. Lewis, On exposed and farthest points in normed linear spaces,J. Austral. Math. Soc.12 (1971), 301–308.

[152] G. A. Edgar, Extremal integral representations,J. Functional Analysis23 (1976), 145–161.

[153] , On the Radon-Nikodym property and martingale convergence, Vector spacemeasures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977), II, Lecture Notesin Phys. 77, Springer, Berlin, 1978, pp. 62–76.

[154] , Two integral representations, Measure theory and its applications (Sher-brooke, Que., 1982), Lecture Notes in Math. 1033, Springer, Berlin, 1983, pp. 193–198.

[155] D. A. Edwards, Separation des fonctions reelles definies sur un simplexe de Choquet,C. R. Acad. Sci. Paris261 (1965), 2798–2800.

[156] , A class of Choquet boundaries that are Baire spaces,Quart. J. Math. OxfordSer. (2)17 (1966), 282–284.

[157] , Minimum-stable wedges of semicontinuous functions,Math. Scand.19(1966), 15–26.

[158] , On uniform approximation of affine functions on a compact convex set.,Quart. J. Math. Oxford Ser. (2)20 (1969), 139–142.

[159] , Systemes projectifs d’ensembles convexes compacts,Bull. Soc. Math. France103 (1975), 225–240.

[160] D. A. Edwards, O. F. K. Kalenda and J. Spurny, A note on intersections of simplices,preprint.

680 Bibliography

[161] D. A. Edwards and G. Vincent-Smith, A Weierstrass-Stone theorem for Choquet sim-plexes,Ann. Inst. Fourier (Grenoble)18 (1968), 261–282, vi.

[162] E. G. Effros, Structure in simplexes,Acta Math.117 (1967), 103–121.

[163] , Structure in simplexes. II,J. Functional Analysis1 (1967), 379–391.

[164] , On a class of real Banach spaces,Israel J. Math.9 (1971), 430–458.

[165] E. G. Effros and J. L. Kazdan, Applications of Choquet simplexes to ellipticand par-abolic boundary value problems,J. Differential Equations8 (1970), 95–134.

[166] , On the Dirichlet problem for the heat equation,Indiana Univ. Math. J.20(1970/1971), 683–693.

[167] A. J. Ellis and A. K. Roy, Dilated sets and characterizations of simplexes,Invent. Math.56 (1980), 101–108.

[168] A. J. Ellis and W. S. So, Isometries and the complex state spaces of uniform algebras,Math. Z.195 (1987), 119–125.

[169] R. Engelking,General topology, PWN—Polish Scientific Publishers, Warsaw, 1977,Translated from the Polish by the author, Monografie Matematyczne, Tom 60. [Mathe-matical Monographs, Vol. 60].

[170] G. C. Evans, Applications of Poincare’s sweeping and process,Proc. Nat. Acad. Sci.19 (1933), 457–461.

[171] , Potentials and positively infinite singularities of harmonic functions,Monatsh.Math. Phys.43 (1936), 419–424.

[172] L. C. Evans and R. F. Gariepy, Wiener’s criterion for the heat equation,Arch. RationalMech. Anal.78 (1982), 293–314.

[173] M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucıa, J. Pelant and V. Zizler,Functional analysis and infinite-dimensional geometry, CMS Books in Mathemat-ics/Ouvrages de Mathematiques de la SMC, 8, Springer-Verlag, New York, 2001.

[174] H. Fakhoury,Une caracterisation des simplexes compacts et des cones reticules.Applications, Seminaire Choquet: 1969/70, Initiationa l’Analyse, Fasc. 1, Exp. 2,Secretariat mathematique, Paris, 1970, p. 12.

[175] , Deux proprietes des simplexes dont l’ensemble des points extremaux estK-analytique,Bull. Sci. Math. (2)95 (1971), 267–272.

[176] K. Floret,Weakly compact sets, Lecture Notes in Mathematics 801, Springer, Berlin,1980, Lectures held at S.U.N.Y., Buffalo, in Spring 1978.

[177] G. B. Folland,A course in abstract harmonic analysis, Studies in Advanced Mathe-matics, CRC Press, Boca Raton, FL, 1995.

[178] V. P. Fonf and J. Lindenstrauss, Boundaries and generation of convex sets,Israel J.Math.136 (2003), 157–172.

[179] V. P. Fonf, J. Lindenstrauss and R. R. Phelps,Infinite dimensional convexity, Handbookof the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 599–670.

Bibliography 681

[180] D. H. Fremlin, Measure-additive coverings and measurable selectors, DissertationesMath. (Rozprawy Mat.)260 (1987), 116.

[181] , Measure theory. Vol. 2, Torres Fremlin, Colchester, 2003, Broad foundations,Corrected second printing of the 2001 original.

[182] , Measure theory. Vol. 1, Torres Fremlin, Colchester, 2004, The irreducibleminimum, Corrected third printing of the 2000 original.

[183] , Measure theory. Vol. 4, Torres Fremlin, Colchester, 2006, Topological mea-sure spaces. Part I, II, Corrected second printing of the 2003 original.

[184] , Cech–analytic spaces,http://www.essex.ac.uk/maths/staff/fremlin/index.shtm(8.12. 1980).

[185] D. H. Fremlin and J. D. Pryce, Semi-extremal sets and measure representations,Proc.London Math. Soc. (3)29 (1974), 502–520.

[186] Z. Frolık, Analytic and Borelian sets in general spaces,Proc. London Math. Soc. (3)21(1970), 674–692.

[187] , A survey of separable descriptive theory of sets and spaces,CzechoslovakMath. J.20 (95) (1970), 406–467.

[188] O. Frostman, Potentiel d’equilibre et capacite des ensembles avec quelques applicationsa la theorie des fonctions,Medd. Lunds Univ. Mat. Sem.3 (1935), 1–118.

[189] , Sur le balayage des mesures,Acta Sci. Math. (Szeged)3 (1938–40), 43–51.

[190] B. Fuglede,Finely harmonic functions, Lecture Notes in Mathematics, Vol. 289,Springer-Verlag, Berlin, 1972.

[191] , Fine potential theory, Potential theory—surveys and problems (Prague, 1987),Lecture Notes in Math. 1344, Springer, Berlin, 1988, pp. 81–97.

[192] I. Gasparis, On contractively complemented subspaces of separableL1-preduals,IsraelJ. Math.128 (2002), 77–92.

[193] D. Gilbarg and N. S. Trudinger,Elliptic partial differential equations of second order,second ed, Grundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences] 224, Springer-Verlag, Berlin, 1983.

[194] E. Glasner and B. Weiss, Kazhdan’s property T and the geometry of the collection ofinvariant measures,Geom. Funct. Anal.7 (1997), 917–935.

[195] A. Gleit, On the structure topology of simplex spaces,Pacific J. Math.34 (1970), 389–405.

[196] , Topologies on the extreme points of compact convex sets,Math. Scand.31(1972), 209–219.

[197] , On the construction of split-face topologies,Trans. Amer. Math. Soc.194(1974), 291–299.

[198] G. Godefroy, Boundaries of a convex set and interpolation sets,Math. Ann.277 (1987),173–184.

682 Bibliography

[199] , Some applications of Simons’ inequality,Serdica Math. J.26 (2000), 59–78.

[200] A. Goullet de Rugy,Geometrie des simplexes, Centre de Documentation Universitaireet S.E.D.E.S. Reunis, Paris, 1968.

[201] A. Goullet de Rugy, C. Schol-Cancelier and B. Taylor-MacGibbon,Quelques resultatsnouveaux sur les points extremaux d’un simplexe compact, Seminaire Choquet, 10eannee (1970/71), Initiationa l’analyse, Fasc. 2, Exp. No. 18, Secretariat Mathematique,Paris, 1971, p. 13.

[202] M. W. Grossman, A Choquet boundary for the product of two compact spaces,Proc.Amer. Math. Soc.16 (1965), 967–971.

[203] , Limits and colimits in certain categories of spaces of continuous functions,Dissertationes Math. Rozprawy Mat.79 (1970), 36.

[204] , Facial quotients of Bauer simplexes,J. London Math. Soc. (2)11 (1975),377–380.

[205] M. Hall, Jr.,Combinatorial theory, second ed, Wiley-Interscience Series in DiscreteMathematics, John Wiley & Sons Inc., New York, 1986, A Wiley-Interscience Publi-cation.

[206] R. W. Hansell, On the nonseparable theory of Borel and Souslin sets,Bull. Amer. Math.Soc.78 (1972), 236–241.

[207] , Hereditarily additive families in descriptive set theory and Borel measurablemultimaps,Trans. Amer. Math. Soc.278 (1983), 725–749.

[208] , A measurable selection and representation theorem in nonseparable spaces,Measure theory, Oberwolfach 1983 (Oberwolfach, 1983), Lecture Notes in Math. 1089,Springer, Berlin, 1984, pp. 86–94.

[209] , Descriptive topology, Recent progress in general topology (Prague, 1991),North-Holland, Amsterdam, 1992, pp. 275–315.

[210] , Descriptive sets and the topology of nonseparable Banach spaces,SerdicaMath. J.27 (2001), 1–66.

[211] A. B. Hansen and Y. Sternfeld, On the characterization of the dimension ofa compactmetric spaceK by the representing matrices ofC(K), Israel J. Math.22 (1975), 148–167.

[212] W. Hansen, Fegen und Dunnheit mit Anwendungen auf die Laplace- undWarmeleitungsgleichung,Ann. Inst. Fourier (Grenoble)21 (1971), 79–121.

[213] , On the identity of Keldych solutions,Czechoslovak Math. J.35(110) (1985),632–638.

[214] W. Hansen and I. Netuka, Continuity properties of concave functions in potential the-ory, J. Convex Anal.15 (2008), 39–53.

[215] P. Harmand andA. Lima, Banach spaces which areM -ideals in their biduals,Trans.Amer. Math. Soc.283 (1984), 253–264.

Bibliography 683

[216] P. Harmand, D. Werner and W. Werner,M -ideals in Banach spaces and Banach alge-bras, Lecture Notes in Mathematics 1547, Springer-Verlag, Berlin, 1993.

[217] F. Hausdorff,Set theory, Second edition. Translated from the German by John R. Au-mann et al, Chelsea Publishing Co., New York, 1962.

[218] R. Haydon, A new proof that every Polish space is the extreme boundary of a simplex,Bull. London Math. Soc.7 (1975), 97–100.

[219] , An extreme point criterion for separability of a dual Banach space, and a newproof of a theorem of Corson,Quart. J. Math. Oxford (2)27 (1976), 379–385.

[220] , Some more characterizations of Banach spaces containingl1, Math. Proc.Cambridge Philos. Soc.80 (1976), 269–276.

[221] J. Heinonen, T. Kilpelainen and O. Martio,Nonlinear potential theory of degenerateelliptic equations, Oxford Mathematical Monographs, The Clarendon Press OxfordUniversity Press, New York, 1993, Oxford Science Publications.

[222] L. L. Helms,Introduction to potential theory, Robert E. Krieger Publishing Co., Hunt-ington, N.Y., 1975, Reprint of the 1969 edition, Pure and Applied Mathematics, Vol.XXII.

[223] M. Herve, Sur les representations integralesa l’aide des points extremaux dans unensemble compact convexe metrisable,C. R. Acad. Sci. Paris253 (1961), 366–368.

[224] H. U. Hess, On a theorem of Cambern,Proc. Amer. Math. Soc.71 (1978), 204–206.

[225] E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer.Math. Soc.80 (1955), 470–501.

[226] B. Hirsberg,A measure theoretic characterization of parallel and split faces and theirconnections with function spaces and algebras, Various Publications Series, No. 16,Matematisk Institut, Aarhus Universitet, Aarhus, 1970.

[227] J. Hoffmann-Jørgensen,Probability in Banach space, Ecole d’Ete de Probabilites deSaint-Flour, VI-1976, Springer-Verlag, Berlin, 1977, pp. 1–186. Lecture Notes inMath., Vol. 598.

[228] W. Hoh and N. Jacob, On the potential theory of the Kolmogorov equation,Math.Nachr.154 (1991), 51–66.

[229] P. Holicky, Cech analytic and almostK-descriptive spaces,Czechoslovak Math. J.43(118) (1993), 451–466.

[230] , Luzin theorems for scattered-K-analytic spaces and Borel measures on them,Atti Sem. Mat. Fis. Univ. Modena44 (1996), 395–413.

[231] , Extensions of Borel measurable maps and ranges of Borel bimeasurable maps,Bull. Pol. Acad. Sci. Math.52 (2004), 151–167.

[232] P. Holicky and T. Keleti, Borel classes of sets of extreme and exposed points inRn,Proc. Amer. Math. Soc.133 (2005), 1851–1859 (electronic).

[233] P. Holicky and V. Komınek, Two remarks on the structure of sets of exposed and ex-treme points,Extracta Math.15 (2000), 547–561.

684 Bibliography

[234] P. Holicky and M. Laczkovich, Descriptive properties of the set of exposed points ofcompact convex sets inR3, Proc. Amer. Math. Soc.132 (2004), 3345–3347 (electronic).

[235] P. Holicky and J. Pelant, Internal descriptions of absolute Borel classes,Topology Appl.141 (2004), 87–104.

[236] P. Holicky and J. Spurny, Perfect images of absolute Souslin and absolute Borel Ty-chonoff spaces,Topology Appl.131 (2003), 281–294.

[237] H. Hollein, A geometrical characterization of Choquet simplexes,Math. Z.160 (1978),249–254.

[238] R. B. Holmes,Geometric functional analysis and its applications, Springer-Verlag,New York, 1975, Graduate Texts in Mathematics, No. 24.

[239] L. Hormander,Notions of convexity, Progress in Mathematics 127, Birkhauser BostonInc., Boston, MA, 1994.

[240] J. Hotta, A remark on regularly convex sets,Kodai Math. Sem. Rep.3 (1951), 37–40,{Volume numbers not printed on issues until Vol.7 (1955).}.

[241] A. Hulanicki and R. R. Phelps, Some applications of tensor products of partially-ordered linear spaces,J. Functional Analysis2 (1968), 177–201.

[242] K. Jacobs,Extremalpunkte konvexer Mengen, Selecta Mathematica, III, Springer,Berlin, 1971, pp. 90–118. Heidelberger Taschenbucher, 86.

[243] R. C. James, Weak compactness and reflexivity,Israel J. Math.2 (1964), 101–119.

[244] , Weakly compact sets,Trans. Amer. Math. Soc.113 (1964), 129–140.

[245] G. Jameson,Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin, 1970.

[246] J. E. Jayne, Generation ofσ-algebras, Baire sets and descriptive Borel sets,Mathe-matika24 (1977), 241–256.

[247] , Metrization of compact convex sets,Math. Ann.234 (1978), 109–115.

[248] J. E. Jayne and C. A. Rogers, The extremal structure of convex sets,J. FunctionalAnalysis26 (1977), 251–288.

[249] F. Jellett, Homomorphisms and inverse limits of Choquet simplexes,Math. Z. 103(1968), 219–226.

[250] , On affine extensions of continuous functions defined on the extreme boundaryof a Choquet simplex,Quart. J. Math. Oxford Ser. (2)36 (1985), 71–73.

[251] W. B. Johnson and J. Lindenstrauss,Basic concepts in the geometry of Banach spaces,Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001,pp. 1–84.

[252] M. Kacena, Products and projective limits of function spaces,Comment. Math. Univ.Carolin. 49 (2008), 547–578.

[253] M. Kacena and J. Spurny, Affine Baire functions on Choquet simplices, preprint.

Bibliography 685

[254] , Affine images of compact convex sets and maximal measures,Bull. Sci. Math.133 (2009), 493–500.

[255] V. M. Kadets, R. V. Shvidkoy, G. G. Sirotkin and D. Werner, Banach spaces with theDaugavet property,Trans. Amer. Math. Soc.352 (2000), 855–873.

[256] R. V. Kadison, A representation theory for commutative topological algebra,Mem.Amer. Math. Soc.,1951 (1951), 39.

[257] O. F. K. Kalenda, (I)-envelopes of closed convex sets in Banach spaces, Israel J. Math.162 (2007), 157–181.

[258] , (I)-envelopes of unit balls and James’ characterization of reflexivity,StudiaMath.182 (2007), 29–40.

[259] O. F. K. Kalenda and J. Spurny, Extending Baire-one functions on topological spaces,Topology Appl.149 (2005), 195–216.

[260] , Boundaries of compact convex sets and fragmentability,J. Funct. Anal.256(2009), 865–880.

[261] J. Kaniewski and R. Pol, Borel-measurable selectors for compact-valued mappings inthe non-separable case,Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.23(1975), 1043–1050.

[262] A. S. Kechris,Classical descriptive set theory, Graduate Texts in Mathematics 156,Springer-Verlag, New York, 1995.

[263] M. V. Keldych, On the Dirichlet problem (Russian),Dokl. Akad. Nauk SSSR32 (1941),308–309.

[264] , On the solubility and the stability of Dirichlet’s problem,Uspekhi Matem.Nauk8 (1941), 171–231.

[265] O.-H. Keller, Die Homoiomorphie der kompakten konvexen Mengen im HilbertschenRaum,Math. Ann.105 (1931), 748–758.

[266] J. L. Kelley, Note on a theorem of Krein and Milman,J. Osaka Inst. Sci. Tech. Part I.3(1951), 1–2.

[267] O. D. Kellog, Unicite des fonctions harmoniques,C. R. Acad. Sci. Paris187 II (1928),526–527.

[268] S. S. Khurana, Pointwise compactness on extreme points,Proc. Amer. Math. Soc.83(1981), 347–348.

[269] S. King and R. Shiflett,Doubly stochastic operators and the history of Birkhoff ’s prob-lem 111, Stochastic processes and functional analysis, Lecture Notes in Pure and Appl.Math. 238, Dekker, New York, 2004, pp. 411–440.

[270] K. Kivisoo and E. Oja, Extension of Simons’ inequality,Proc. Amer. Math. Soc.133(2005), 3485–3496 (electronic).

[271] V. Klee, Some new results on smoothness and rotundity in normed linear spaces.,Math.Ann.139 (1959), 51–63 (1959).

686 Bibliography

[272] V. L. Klee, Jr., Extremal structure of convex sets. II,Math. Z.69 (1958), 90–104.

[273] J. Kohn, Harmonische Raume mit einer Basis semiregularer Mengen, SeminaruberPotentialtheorie, Springer, Berlin, 1968, pp. 1–12.

[274] , Barycenters of unique maximal measures,J. Functional Analysis6 (1970),76–82.

[275] J. Kohn and M. Sieveking, Regulare und extremale Randpunkte in der Potentialtheorie,Rev. Roumaine Math. Pures Appl.12 (1967), 1489–1502.

[276] J. Kolar and J. Lukes, Simultaneous solutions of the weak Dirichlet problem,PotentialAnal.15 (2001), 17–21, ICPA98 (Hammamet).

[277] P. P. Korovkin, On convergence of linear positive operators in the space of continuousfunctions,Doklady Akad. Nauk SSSR (N.S.)90 (1953), 961–964.

[278] , Linear operators and approximation theory, Translated from the Russian ed.(1959). Russian Monographs and Texts on Advanced Mathematics and Physics, Vol.III, Gordon and Breach Publishers, Inc., New York, 1960.

[279] R. A. Kortram, The extreme points of a class of functions with positive realpart,Bull.Belg. Math. Soc. Simon Stevin4 (1997), 449–459.

[280] G. Koumoullis, A generalization of functions of the first class,Topology Appl.50(1993), 217–239.

[281] M. Kraus, A note on the uniform approximation of continuous affine functions,Expo.Math.27 (2009), 73–78.

[282] U. Krause, Der Satz von Choquet als ein abstrakter Spektralsatz und vice versa,Math.Ann.184 (1970), 275–296.

[283] M. Krein and D. Milman, On extreme points of regular convex sets,Studia Math.9(1940), 133–138.

[284] A. Kufner, O. John and S. Fucık, Function spaces, Noordhoff International Publish-ing, Leyden, 1977, Monographs and Textbooks on Mechanics of Solids and Fluids;Mechanics: Analysis.

[285] C. Kuratowski,Topologie. Vol. I, Monografie Matematyczne, Tom 20, PanstwoweWydawnictwo Naukowe, Warsaw, 1958, 4emeed.

[286] K. Kuratowski and A. Maitra, Some theorems on selectors and their applications tosemi-continuous decompositions,Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom.Phys.22 (1974), 877–881.

[287] K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors,Bull. Acad.Polon. Sci. Ser. Sci. Math. Astronom. Phys.13 (1965), 397–403.

[288] C. De La Vallee Poussin, Sur quelques extensions de la methode du balayage dePoincare et sur le probleme de Dirichlet,C. R. Acad. Sci. Paris192 (1931), 651–653.

[289] , Extension de la methode du balayage de Poincare et probleme de Dirichlet,Ann. Inst. H. Poincare2 (1932), 169–232.

Bibliography 687

[290] H. E. Lacey,The isometric theory of classical Banach spaces, Springer-Verlag, NewYork, 1974, Die Grundlehren der mathematischen Wissenschaften, Band 208.

[291] H. E. Lacey and P. D. Morris, On spaces of typeA(K) and their duals,Proc. Amer.Math. Soc.23 (1969), 151–157.

[292] A. J. Lazar, Affine products of simplexes,Math. Scand.22 (1968), 165–175 (1969).

[293] , Spaces of affine continuous functions on simplexes,Trans. Amer. Math. Soc.134 (1968), 503–525.

[294] , Extreme boundaries of convex bodies inl2, Israel J. Math.20 (1975), 369–374.

[295] A. J. Lazar and J. Lindenstrauss, On Banach spaces whose duals areL1 spaces,IsraelJ. Math.4 (1966), 205–207.

[296] , Banach spaces whose duals areL1 spaces and their representing matrices,Acta Math.126 (1971), 165–193.

[297] H. Lebesgue, Sur le cas d’impossibilite du probleme de Dirichlet ordinaire,C. R.Seances Soc. Math. France17 (1912).

[298] , Conditions de regularite, conditions d’irregularite, conditions d’impossibilitedans le probleme de Dirichlet,C. R. Acad. Sci. Paris178 (1924), 349–354.

[299] C. Leger, Une demonstration du theoreme de A. J. Lazar sur les simplexes compacts,C. R. Acad. Sci. Paris Ser. A-B265 (1967), A830–A831.

[300] A. Liapounoff, Sur les fonctions-vecteurs completement additives,Bull. Acad. Sci.URSS. Ser. Math. [Izvestia Akad. Nauk SSSR]4 (1940), 465–478.

[301] G. M. Lieberman,Second order parabolic differential equations, World Scientific Pub-lishing Co. Inc., River Edge, NJ, 1996.

[302] A. Lima, On continuous convex functions and split faces,Proc. London Math. Soc. (3)25 (1972), 27–40.

[303] H. Lin, An introduction to the classification of amenableC∗-algebras, World ScientificPublishing Co. Inc., River Edge, NJ, 2001.

[304] J. Lindenstrauss, Extension of compact operators,Mem. Amer. Math. Soc. No.48(1964), 112.

[305] , A remark on extreme doubly stochastic measures,Amer. Math. Monthly72(1965), 379–382.

[306] , A short proof of Liapounoff’s convexity theorem,J. Math. Mech.15 (1966),971–972.

[307] J. Lindenstrauss, G. Olsen and Y. Sternfeld, The Poulsen simplex,Ann. Inst. Fourier(Grenoble)28 (1978), vi, 91–114.

[308] J. Lindenstrauss and D. E. Wulbert, On the classification of the Banach spaces whoseduals areL1 spaces,J. Functional Analysis4 (1969), 332–349.

688 Bibliography

[309] Z. Lipecki, On compactness and extreme points of some sets of quasi-measuresandmeasures. III,Manuscripta Math.117 (2005), 463–473.

[310] Z. Lipecki, V. Losert and J. Spurny, Products of simplices, preprint.

[311] L. H. Loomis, Unique direct integral decompositions on convex sets,Amer. J. Math.84(1962), 509–526.

[312] D. H. Luecking and L. A. Rubel,Complex analysis, Universitext, Springer-Verlag, NewYork, 1984, A functional analysis approach.

[313] J. Lukes, Principal solution of the Dirichlet problem in potential theory,Comment.Math. Univ. Carolinae14 (1973), 773–778.

[314] , Theoreme de Keldych dans la theorie axiomatique de Bauer des fonctionsharmoniques,Czechoslovak Math. J.24(99) (1974), 114–125.

[315] , Semiregular sets in harmonic spaces,Casopis Pest. Mat.100 (1975), 195–197.

[316] , The Lusin-Menchoff property of fine topologies,Comment. Math. Univ. Car-olinae18 (1977), 515–530.

[317] J. Lukes and J. Maly, On the boundary behaviour of the Perron generalized solution,Math. Ann.257 (1981), 355–366.

[318] , Fine hyperharmonicity without axiom D,Math. Ann.261 (1982), 299–306.

[319] J. Lukes, J. Maly, I. Netuka, M. Smrcka and J. Spurny, On approximation of affineBaire-one functions,Israel J. Math.134 (2003), 255–287.

[320] J. Lukes, J. Maly and L. Zajıcek,Fine topology methods in real analysis and potentialtheory, Lecture Notes in Mathematics 1189, Springer-Verlag, Berlin, 1986.

[321] J. Lukes, T. Mocek and I. Netuka, Exposed sets in potential theory,Bull. Sci. Math.130(2006), 646–659.

[322] J. Lukes, T. Mocek, M. Smrcka and J. Spurny, Choquet like sets in function spaces,Bull. Sci. Math.127 (2003), 397–437.

[323] J. Lukes and I. Netuka, The Wiener type solution of the Dirichlet problem in potentialtheory,Math. Ann.224 (1976), 173–178.

[324] , What is the right solution of the Dirichlet problem?, Romanian-Finnish Sem-inar on Complex Analysis (Proc., Bucharest, 1976), Lecture Notes in Math. 743,Springer, Berlin, 1979, pp. 564–572.

[325] , Extreme harmonic functions on a ball,Expo. Math.22 (2004), 83–91.

[326] W. Lusky, The Gurarij spaces are unique,Arch. Math. (Basel)27 (1976), 627–635.

[327] , On separable Lindenstrauss spaces,J. Functional Analysis26 (1977), 103–120.

[328] , Separable Lindenstrauss spaces, Functional Analysis: surveys and recent re-sults (Proc. Conf., Paderhorn, 1976), Notas Mat. 63, North-Holland, Amsterdam, 1977,pp. 15–28.

Bibliography 689

[329] , On a construction of Lindenstrauss and Wulbert,J. Funct. Anal.31 (1979),42–51.

[330] , On nonseparable simplex spaces,Math. Scand.61 (1987), 276–285.

[331] , Every L1-predual is complemented in a simplex space,Israel J. Math.64(1988), 169–178.

[332] , Every separableL1-predual is complemented in aC∗-algebra,Studia Math.160 (2004), 103–116.

[333] B. MacGibbon, A criterion for the metrizability of a compact convex set in terms of theset of extreme points,J. Functional Analysis11 (1972), 385–392.

[334] J. Maly and W. P. Ziemer,Fine regularity of solutions of elliptic partial differentialequations, Mathematical Surveys and Monographs 51, American Mathematical Soci-ety, Providence, RI, 1997.

[335] R. D. Mauldin, Baire functions, Borel sets, and ordinary function systems,Advances inMath.12 (1974), 418–450.

[336] J. N. McDonald, Compact convex sets with the equal support property,Pacific J. Math.37 (1971), 429–443.

[337] , Maximal measures and abstract Dirichlet problems,Quart. J. Math. OxfordSer. (2)22 (1971), 239–246.

[338] , Some constructions with Choquet simplexes,J. London Math. Soc. (2)6(1973), 307–310.

[339] P.-A. Meyer, Sur les demonstrations nouvelles du theoreme de Choquet,SeminaireBrelot–Choquet–Deny (Theorie de Potentiel)7 (1961/62), 7.

[340] P. R. Meyer, The Baire order problem for compact spaces,Duke Math. J.33 (1966),33–39.

[341] E. Michael, Continuous selections. I,Ann. of Math. (2)63 (1956), 361–382.

[342] , Selected Selection Theorems,Amer. Math. Monthly63 (1956), 233–238.

[343] , The product of a normal space and a metric space need not be normal,Bull.Amer. Math. Soc.69 (1963), 375–376.

[344] E. Michael and A. Pełczynski, Separable Banach spaces which admitln∞ approxima-

tions,Israel J. Math.4 (1966), 189–198.

[345] D. Mil ′man, Isometry and extremal points,Doklady Akad. Nauk SSSR (N.S.)59 (1948),1241–1244.

[346] D. P. Mil′man, Facial characterization of convex sets; extremal elements,TrudyMoskov. Mat. Obsc. 22 (1970), 63–126.

[347] G. Mokobodzki, Balayage defini par un cone convexe de fonctions numeriques sur unespace compact,C. R. Acad. Sci. Paris254 (1962), 803–805.

690 Bibliography

[348] G. Mokobodzki and D. Sibony,Cones de fonctions et theorie du potentiel. I. Les noy-aux associesa un cone de fonctions, Seminaire de Theorie du Potentiel, dirige par M.Brelot, G. Choquet et J. Deny: 1966/67, Exp. 8, Secretariat mathematique, Paris, 1968,p. 35.

[349] A. F. Monna, Note sur le probleme de Dirichlet,Nieuw Arch. Wisk. (3)19 (1971),58–64.

[350] W. B. Moors and E. A. Reznichenko, Separable subspaces of affine function spacesonconvex compact sets,Topology Appl.155 (2008), 1306–1322.

[351] W. B. Moors and J. Spurny, On the topology of pointwise convergence on the bound-aries ofL1-preduals,Proc. Amer. Math. Soc.137 (2009), 1421–1429.

[352] I. Namioka and R. R. Phelps, Tensor products of compact convex sets,Pacific J. Math.31 (1969), 469–480.

[353] I. Namioka and R. Pol,σ-fragmentability and analyticity,Mathematika43 (1996), 172–181.

[354] M. A. Navarro, Some characterizations of finite-dimensional Hilbert spaces,J. Math.Anal. Appl.223 (1998), 364–365.

[355] I. Netuka, A remark on semiregular sets,Casopis Pest. Mat.98 (1973), 419–421.

[356] , Thinness and the heat equation,Casopis Pest. Mat.99 (1974), 293–299.

[357] , The classical Dirichlet problem and its generalizations, Potential theory,Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), Lecture Notes in Math. 787,Springer, Berlin, 1980, pp. 235–266.

[358] , The Dirichlet problem for harmonic functions,Amer. Math. Monthly87(1980), 621–628.

[359] , Extensions of operators and the Dirichlet problem in potential theory, in:Pro-ceedings of the 13th winter school on abstract analysis (Srnı, 1985), 10, pp. 143–163(1986), 1985.

[360] , The Ninomiya operators and the generalized Dirichlet problem in potentialtheory,Osaka J. Math.23 (1986), 741–750.

[361] , Separation properties involving harmonic functions,Expo. Math.18 (2000),333–337.

[362] , The work of Heinz Bauer in potential theory, Selecta, de Gruyter, Berlin,2003, pp. 29–41.

[363] I. Netuka and J. Vesely, On harmonic functions (solution of the problem 6393 [1982,502] proposed by G.A. Edgar),Amer. Math. Monthly91 (1984), 61–62.

[364] G. Nobeling and H. Bauer, Allgemeine Approximationskriterien mit Anwendungen,Jber. Deutsch. Math. Verein.58 (1955), 54–72.

[365] R. C. O’Brien, On the openness of the barycentre map,Math. Ann.223 (1976), 207–212.

Bibliography 691

[366] E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banachspaces containingl1, Israel J. Math.20 (1975), 375–384.

[367] E. Oja, A proof of the Simons inequality,Acta Comment. Univ. Tartu. Math.(1998),27–28.

[368] S. Papadopoulou, On the geometry of stable compact convex sets,Math. Ann.229(1977), 193–200.

[369] , Stabile konvexe Mengen,Jahresber. Deutsch. Math.-Verein.84 (1982), 92–106.

[370] F. Perdrizet, Espaces de Banach ordonnes et ideaux,C. R. Acad. Sci. Paris Ser. A-B269(1969), A393–A396.

[371] , Espaces de Banach ordonnes et ideaux,J. Math. Pures Appl. (9)49 (1970),61–98.

[372] O. Perron, Eine neue Behandlung der ersten Randwertaufgabe fur ∆u = 0, Math. Z.18(1923), 42–54.

[373] H. Pfitzner,Boundaries for Banach spaces determine weak compactness, preprint.

[374] R. R. Phelps,Lectures on Choquet’s theorem, second ed, Lecture Notes in Mathematics1757, Springer-Verlag, Berlin, 2001.

[375] H. Poincare, Sur les Equations aux Derivees Partielles de la Physique Mathematique,Amer. J. Math.12 (1890), 211–294.

[376] E. T. Poulsen, A simplex with dense extreme points,Ann. Inst. Fourier. Grenoble11(1961), 83–87, XIV.

[377] G. B. Price, On the extreme points of convex sets,Duke Math. J.3 (1937), 56–67.

[378] J. D. Pryce, On the representation and some separation properties of semi-extremalsubsets of convex sets,Quart. J. Math. Oxford Ser. (2)20 (1969), 367–382.

[379] , A device of R. J. Whitley’s applied to pointwise compactness in spaces ofcontinuous functions,Proc. London Math. Soc. (3)23 (1971), 532–546.

[380] V. Ptak,A combinatorial lemma on the existence of convex means and its application toweak compactness, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence,R.I., 1963, pp. 437–450.

[381] M. Raja, On some class of Borel measurable maps and absolute Borel topologicalspaces,Topology Appl.123 (2002), 267–282.

[382] , First Borel class sets in Banach spaces and the asymptotic-norming property,Israel J. Math.138 (2003), 253–270.

[383] R. M. Rakestraw, A representation theorem for real convex functions,Pacific J. Math.60 (1975), 165–168.

[384] M. Rao, Measurable selections of representing measures,Quart. J. Math. Oxford Ser.(2) 22 (1971), 571–573.

[385] T. S. S. R. K. Rao, Isometries ofAC(K), Proc. Amer. Math. Soc.85 (1982), 544–546.

692 Bibliography

[386] R. Remak,Uber potentialkonvexe Funktionen,Math. Z.20 (1924), 126–130.

[387] E. A. Reznichenko, Convex compact spaces and their maps,Topology Appl.36(1990), 117–141, Seminar on General Topology and Topological Algebra (Moscow,1988/1989).

[388] A. W. Roberts and D. E. Varberg,Convex functions, Academic Press [A subsidiary ofHarcourt Brace Jovanovich, Publishers], New York-London, 1973, Pure and AppliedMathematics, Vol. 57.

[389] M. S. Robertson, On the coefficients of a typically-real function,Bull. Amer. Math. Soc.41 (1935), 565–572.

[390] G. Rode, Superkonvexitat und schwache Kompaktheit,Arch. Math. (Basel)36 (1981),62–72.

[391] M. Rogalski,Espaces de Banach ordonnes, simplexes, frontieres deSilov et problemede Dirichlet, Seminaire Choquet: 1965/66, Initiationa l’ Analyse, Fasc. 2, Exp. 12,Secretariat mathematique, Paris, 1968, p. 62.

[392] , Operateurs de Lion, projecteurs boeliens et simplexes analytiques,J. Func-tional Analysis2 (1968), 458–488.

[393] , Representations fonctionnelles d’espaces vectoriels reticules, Seminaire Cho-quet: 1965/66, Initiationa l’ Analyse, Fasc. 1, Exp. 2, Secretariat mathematique, Paris,1968, p. 31.

[394] C. A. Rogers and J. E. Jayne,K-analytic sets, Analytic sets, Academic Press Inc.[Harcourt Brace Jovanovich Publishers], London, 1980, pp. 1–181.

[395] M. Rørdam,Classification of nuclear, simpleC∗-algebras, Classification of nuclearC∗-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci. 126, Springer,Berlin, 2002, pp. 1–145.

[396] M. Rørdam and E. Størmer,Classification of nuclearC∗-algebras. Entropy in operatoralgebras, Encyclopaedia of Mathematical Sciences 126, Springer-Verlag, Berlin, 2002,Operator Algebras and Non-commutative Geometry, 7.

[397] H. Rosenthal,On the Choquet representation theorem, Functional analysis (Austin,TX, 1986–87), Lecture Notes in Math. 1332, Springer, Berlin, 1988, pp. 1–32.

[398] , A characterization of Banach spaces containingc0, J. Amer. Math. Soc.7(1994), 707–748.

[399] A. K. Roy, Closures of faces of compact convex sets,Ann. Inst. Fourier (Grenoble)25(1975), 221–234.

[400] , Errata: “Closures of faces of compact convex sets” (Ann. Inst. Fourier (Greno-ble)25 (1975), no. 2, 221–234),Ann. Inst. Fourier (Grenoble)26 (1976).

[401] N. M. Roy, Extreme points of convex sets in infinite-dimensional spaces,Amer. Math.Monthly94 (1987), 409–422.

[402] W. Rudin,Real and complex analysis, third ed, McGraw-Hill Book Co., New York,1987.

Bibliography 693

[403] , Functional analysis, second ed, International Series in Pure and AppliedMathematics, McGraw-Hill Inc., New York, 1991.

[404] M. Ruiz Galan and S. Simons, A new minimax theorem and a perturbed James’s theo-rem,Bull. Austral. Math. Soc.66 (2002), 43–56.

[405] J. J. Saccoman, On the origin of extreme points and the Kreın-Mil ′man theorem,Aus-tral. Math. Soc. Gaz.17 (1990), 10–12.

[406] J. Saint-Raymond, Fonctions boreliennes sur un quotient,Bull. Sci. Math. (2)100(1976), 141–147.

[407] , Fonctions convexes sur un convexe borne complet,Bull. Sci. Math. (2)102(1978), 331–336.

[408] , Fonctions convexes de premiere classe,Math. Scand.54 (1984), 121–129.

[409] H. H. Schaefer,Topological vector spaces, The Macmillan Co., New York, 1966.

[410] H. Schirmeier and U. Schirmeier, Einige Bemerkungenuber den Satz von Keldych,Math. Ann.236 (1978), 245–254.

[411] G. Schober,Univalent functions—selected topics, Lecture Notes in Mathematics, Vol.478, Springer-Verlag, Berlin, 1975.

[412] Z. Semadeni, Free compact convex sets,Bull. Acad. Polon. Sci. Ser. Sci. Math. As-tronom. Phys.13 (1965), 141–146.

[413] , Categorical methods in convexity, Proc. Colloquium on Convexity (Copen-hagen, 1965), Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 281–307.

[414] , Banach spaces of continuous functions. Vol. I, PWN—Polish Scientific Pub-lishers, Warsaw, 1971, Monografie Matematyczne, Tom 55.

[415] D. Sibony,Cones des fonctions et potentiels, Cours de 3eme Cycle, Fac. des Sci. deParis, mimeographed, 1967–68, p. 150.

[416] W. Sierpinski, Les fonctions continues et la propriete de Baire,Fund. Math.28 (1937),120–121.

[417] S. Simons, A convergence theorem with boundary,Pacific J. Math.40 (1972), 703–708.

[418] , An eigenvector proof of Fatou’s lemma for continuous functions,Math. Intel-ligencer17 (1995), 67–70.

[419] R. Sine, Geometric theory of a single Markov operator,Pacific J. Math.27 (1968),155–166.

[420] J. Spurny, Baire classes of Banach spaces and strongly affine functions,Trans. Amer.Math. Soc., to appear.

[421] , Borel sets and functions in topological spaces, preprint.

[422] , Boundary problem forL1–preduals,Illinois J. Math., to appear.

[423] , On the Dirichlet problem of extreme points for non-continuous functions,Israel J. Math., to appear.

694 Bibliography

[424] , On the Dirichlet problem for functions of the first Baire class,Comment. Math.Univ. Carolin.42 (2001), 721–728.

[425] , Representation of abstract affine functions,Real Anal. Exchange28(2002/03), 337–354.

[426] , The Dirichlet problem for Baire-one functions,Cent. Eur. J. Math.2 (2004),260–271 (electronic).

[427] , Affine Baire-one functions on Choquet simplexes,Bull. Austral. Math. Soc.71 (2005), 235–258.

[428] , The weak Dirichlet problem for Baire functions,Proc. Amer. Math. Soc.134(2006), 3153–3157 (electronic).

[429] , Complementability of spaces of harmonic functions,Potential Anal. 29(2008), 271–302.

[430] , Automatic boundedness of affine functions,Houston J. Math.35 (2009), 553–561.

[431] , The Dirichlet problem for Baire–two functions on simplices,Bull. Austral.Math. Soc.79 (2009), 285–297.

[432] J. Spurny and O. F. K. Kalenda, A solution of the abstract Dirichlet problem for Baire-one functions,J. Funct. Anal.232 (2006), 259–294.

[433] J. Spurny and M. Zeleny, Additive families of low Borel classes and Borel measurableselectors,Canad. Math. Bull., to appear.

[434] S. M. Srivastava,A course on Borel sets, Graduate Texts in Mathematics 180, Springer-Verlag, New York, 1998.

[435] P. J. Stacey, Split faces and projective sets in a metrizable compact convex set,Math.Ann.219 (1976), 167–170.

[436] , Choquet simplices with prescribed extreme andSilov boundaries,Quart. J.Math. Oxford Ser. (2)30 (1979), 469–482.

[437] S. Sternberg,Lectures on differential geometry, second ed, Chelsea Publishing Co.,New York, 1983, With an appendix by Sternberg and Victor W. Guillemin.

[438] Y. Sternfeld,Characterization of Bauer simplices and some other classes of Choquetsimplices by their representing matrices, Notes in Banach spaces, Univ. Texas Press,Austin, Tex., 1980, pp. 306–358.

[439] A. H. Stone, Non-separable Borel sets,Rozprawy Mat.28 (1962), 41.

[440] E. Størmer, On partially ordered vector spaces and their duals, with applications tosimplexes andC∗-algebras,Proc. London Math. Soc. (3)18 (1968), 245–265.

[441] S. Straszewicz,Uber exponierte Punkte abgeschlossener Punktmengen,Fund. Math.24 (1935), 139–143.

[442] M. Talagrand, Les fonctions affines sur[0, 1]N ayant la propriete de Baire faible sontcontinues,Seminaire Choquet (Initiation a l’analyse)15 (1975/76).

Bibliography 695

[443] , Selection mesurable de mesures maximales simpliciales,Bull. Sci. Math. (2)102 (1978), 49–56.

[444] , Deux generalisations d’un theoreme de I. Namioka,Pacific J. Math.81 (1979),239–251.

[445] , Sur les convexes compacts dont l’ensemble des points extremaux estK-analytique,Bull. Soc. Math. France107 (1979), 49–53.

[446] , Three convex sets,Proc. Amer. Math. Soc.89 (1983), 601–607.

[447] , A new type of affine Borel function,Math. Scand.54 (1984), 183–188.

[448] , Choquet simplexes whose set of extreme points isK-analytic, Ann. Inst.Fourier (Grenoble)35 (1985), 195–206.

[449] S. Teleman, An introduction to Choquet theory with the applications to reductionthe-ory, Increst Preprint Series in Mathematics(1980), 1–294.

[450] , Measure–theoretic properties of the Choquet and maximal topologies,IncrestPreprint Series in Mathematics(1982), 1–41.

[451] , On the regularity of the boundary measures, Complex analysis—fifthRomanian-Finnish seminar, Part 2 (Bucharest, 1981), Lecture Notes in Math. 1014,Springer, Berlin, 1983, pp. 296–315.

[452] , Topological properties of the boundary measures, Studies in probability andrelated topics, Nagard, Rome, 1983, pp. 457–463.

[453] , Measure–theoretic properties of the maximal orthogonal topology,IncrestPreprint Series in Mathematics(1984), 1–35.

[454] , On the boundary barycentric calculus,J. Funct. Anal.78 (1988), 85–98.

[455] , Sur les mesures maximales,C. R. Acad. Sci. Paris Ser. I Math. 318 (1994),525–528.

[456] F. Topsøe and J. Hoffmann-Jørgensen,Analytic spaces and their applications, Ana-lytic sets, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1980,pp. 317–401.

[457] F. Vasilesco, Sur les singularites des fonctions harmoniques,J. Math. Pures Appl.9(1930), 81–111.

[458] , La notion de point irregulier dans le probleme de Dirichlet, Hermann, Paris,1938, Actualites Scientifiques et Industrielles, no. 660, 61 pp.

[459] J. Vesely, Sequence solutions of the Dirichlet problem,Casopis Pest. Mat.106 (1981),84–93, 102, With a loose Russian summary.

[460] J. Vesterstrøm, On open maps, compact convex sets, and operator algebras, J. LondonMath. Soc. (2)6 (1973), 289–297.

[461] G. F. Vincent-Smith, Uniform approximation of harmonic functions,Ann. Inst. Fourier(Grenoble)19 (1969), 339–353 (1970).

696 Bibliography

[462] H. von Weizsacker, Der Satz von Choquet-Bishop-de Leeuw fur konvexe nicht kom-pakte Mengen straffer Maßeuber beliebigen Grundraumen,Math. Z.142 (1975), 161–165.

[463] , A note on infinite dimensional convex sets,Math. Scand.38 (1976), 321–324.

[464] H. von Weizsacker and G. Winkler,Noncompact extremal integral representations:some probabilistic aspects, Functional analysis: surveys and recent results, II (Proc.Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), Notas Mat. 68,North-Holland, Amsterdam, 1980, pp. 115–148.

[465] D. Werner,Some lifting theorems for bounded linear operators, Functional analysis(Essen, 1991), Lecture Notes in Pure and Appl. Math. 150, Dekker, New York, 1994,pp. 279–291.

[466] , Recent progress on the Daugavet property,Irish Math. Soc. Bull.(2001), 77–97.

[467] D. V. Widder,The Laplace Transform, Princeton Mathematical Series, v. 6, PrincetonUniversity Press, Princeton, N. J., 1941.

[468] N. Wiener, Certain notions in potential theory,J. Math. Mass.3 (1924), 24–51.

[469] , The Dirichlet problem,J. Math. Mass.3 (1924), 127–146.

[470] , Note on a paper of O. Perron,J. Math. Mass.4 (1925), 21–32.

[471] S. Willard, Some examples in the theory of Borel sets,Fund. Math.71 (1971), 187–191.

[472] W. Wils, The ideal center of partially ordered vector spaces,Acta Math.127 (1971),41–77.

[473] G. Winkler, Choquet order and simplices with applications in probabilistic models,Lecture Notes in Mathematics 1145, Springer-Verlag, Berlin, 1985.

[474] R. Wittmann, On the existence of Shilov boundaries,Proc. Amer. Math. Soc.89 (1983),62–64.

[475] , Shilov points and Shilov boundaries,Math. Ann.263 (1983), 237–250.

[476] J. D. Maitland Wright, On approximating concave functions by convex functions,Bull.London Math. Soc.5 (1973), 221–222.

[477] K. Yosida and M. Fukamiya, On regularly convex sets,Proc. Imp. Acad. Tokyo17(1941), 49–52.

[478] S. Zaremba, Sur le principe de Dirichlet,Acta Math.34 (1911), 293–316.

[479] M. Zippin, On some subspaces of Banach spaces whose duals areL1 spaces,Proc.Amer. Math. Soc.23 (1969), 378–385.