A short proof of the existence of master test plans

TL;DR

This paper provides a simple and elementary proof demonstrating that in certain metric measure spaces, a specific test plan can identify the minimal weak upper gradient of all Sobolev functions, simplifying existing theoretical understanding.

Contribution

It offers a new, straightforward proof of a known fact about the existence of a universal test plan in separable Sobolev spaces on metric measure spaces.

Findings

Existence of a sufficient test plan in separable Sobolev spaces

Elementary proof simplifies previous complex arguments

Applicable to metric measure spaces with separable Sobolev spaces

Abstract

The aim of this brief note is to provide a quick and elementary proof of the following known fact: on a metric measure space whose Sobolev space is separable, there exists a test plan that is sufficient to identify the minimal weak upper gradient of every Sobolev function.