On master test plans for the space of BV functions

TL;DR

This paper demonstrates that in metric measure spaces, a countable set of test plans can recover all BV functions and their total variations, with specific conditions in non-branching ${ m CD}(K,N)$ spaces.

Contribution

It establishes the sufficiency of a countable collection of test plans for BV functions in general metric measure spaces, and specifies geodesic concentration in non-branching ${ m CD}(K,N)$ spaces.

Findings

Countable test plans recover all BV functions and total variations.

In non-branching ${ m CD}(K,N)$ spaces, test plans can be concentrated on geodesics.

The results apply to arbitrary metric measure spaces and specific structured spaces.

Abstract

We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all $\rm BV$ functions and their total variation measures. In the setting of non-branching ${\sf CD}(K,N)$ spaces (with finite reference measure), we can additionally require these test plans to be concentrated on geodesics.