The translation invariant product measure problem in non-sigma finite case

TL;DR

This paper presents an example of a non-translation invariant product measure derived from two translation-invariant measures, highlighting the potential for infinitely many such measures when sigma-finiteness is not assumed.

Contribution

It provides the first explicit example of a non-sigma finite, non-translation invariant product measure, expanding understanding of measure theory.

Findings

Existence of non-translation invariant product measures from sigma-finite measures

Infinite variety of product measures possible without sigma-finiteness

Implications for measure theory and invariance properties

Abstract

We give an example of non-translation invariant product measure obtained from two translation invariant measures, one of which is non-sigma finite. This particular example also suggests that there can be infinitely many product measures if we abandon the sigma-finiteness assumption.