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 Rn is closely related to the Lebesgue measure on Rn. We show that the Ashtekar-Lewandowski measure on the space of generalized G-connections for a compact, connected, 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, under very mild conditions, the Ashtekar-Lewandowski measure is supported on nowhere-continuous generalized connections. © 2014 AIP Publishing LLC.