My Apologies

Joint work with Gui-Qiang Chen, Monica Torres

The Problem

The answer is Yes – Main Result

When F is assumed to be only bounded, measurableWe can’t expect the same answer

What measures are admissible?

One of the Fundamental results of GMT

Sets of Finite Perimeter

The Density Topology

The Density Topology – The Union of uncountably many measurable sets Is still measurable

The measure-theoretic normal to an arbitrary measurable set

The Set E lies on one side of the plane determined by the normal

The Reduced Boundary of E

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A Pathological Example

More of Example – Swiss Cheese

Example of F in 3-space

Fuglede: The notion of “almost every set”

Ties to potential theory and Complex analysis

A Slight Diversion – The Coarea Formula

Fuglede’s result in a more meaningful setting

The Normal Trace of F on an arbitrary set

Preliminary result

More of the Hypothesis

Proof of the Result

Consider fields F that are bounded and for which the divergence is a signed measure

The Divergence Theorem for “almost every set” Fuglede

Proof of Fuglede’s Result – Quick Sketch

The Main Problem

Definition of Normal Trace