On a Gel'fand-Yaglom-Peres theorem for f-divergences

TL;DR

This paper generalizes a classical result by showing that the f-divergence between two probability measures can be approximated by the supremum over all finite measurable partitions, extending previous work on specific divergences.

Contribution

It establishes a Gelfand-Yaglom-Peres type theorem for f-divergences, broadening the understanding of divergence measures in probability theory.

Findings

f-divergence equals the supremum over finite measurable partitions

Generalizes previous results for specific divergences

Extends classical theorems to a broader class of divergence measures

Abstract

It is shown that the $f$-divergence between two probability measures $P$ and $R$ equals the supremum of the same $f$-divergence computed over all finite measurable partitions of the original space, thus generalizing results previously proved by Gel'fand and Yaglom and by Peres for the Information Divergence and more recently by Dukkipati, Bhatnagar and Murty for the Tsallis' and Renyi's divergences.