On the Ashtekar-Lewandowski Measure as a Restriction of the Product One

Abstract

It is known that the $k$-dimensional Hausdorff measure on a $k$-dimensional submanifold of ${\mathbb{R}}^n$ is closely related to the Lebesgue measure on ${\mathbb{R}}^n$. We show that the Ashtekar-Lewandowski measure on the space of generalized $G$-connections for a compact, semi-simple, Lie group $G$, is analogously related to the product measure on the set of all $G$-valued functions on the group of loops. We also show that, the Ashtekar-Lewandowski measure is, under very mild conditions, supported on nowhere-continuous generalized connections.

11 pages, RevTeX, several explanations clarified and expanded with no change to the results, 3 references added, typos fixed

References in corpus (1)

• On the Holonomic Equivalence of Two Curves

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