An uncountable Moore-Schmidt theorem

TL;DR

This paper extends the Moore-Schmidt theorem to broader contexts involving arbitrary groups, measure spaces, and abelian groups, using a novel duality approach for measurable maps.

Contribution

It introduces a generalized version of the Moore-Schmidt theorem applicable to uncountable groups and new duality techniques for measurable functions.

Findings

Extended Moore-Schmidt theorem to uncountable groups

Developed a conditional Pontryagin duality for measurable maps

Proved triviality of first cohomology in broader settings

Abstract

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a "conditional" Pontryagin duality for spaces of abstract measurable maps.