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.