Boundedness of precompact sets of metric measure spaces

TL;DR

This paper provides a detailed proof of Gromov's claim that precompact sets of metric measure spaces are bounded under the box distance and Lipschitz order, clarifying foundational aspects of metric measure space theory.

Contribution

It offers a comprehensive proof of Gromov's statement regarding the boundedness of precompact metric measure spaces in specific metric frameworks.

Findings

Precompact sets are bounded in the box distance.

Precompact sets are bounded with respect to the Lipschitz order.

Provides detailed proof of a key theoretical statement.

Abstract

We give a detailed proof to Gromov's statement that precompact sets of metric measure spaces are bounded with respect to the box distance and the Lipschitz order.