Boundedness of measured Gromov-Hausdorff precompact sets of metric measure spaces in pyramids

TL;DR

This paper proves that precompact sets of metric measure spaces within a pyramid are bounded by a specific space, advancing the understanding of Gromov-Hausdorff convergence and metric measure space structure.

Contribution

It establishes boundedness of precompact sets in pyramids, extending Gromov's results with detailed proofs and related findings.

Findings

Precompact sets in pyramids are bounded by a metric measure space.

The work extends Gromov's boundedness results.

Several related properties of metric measure spaces are derived.

Abstract

We prove that any measured Gromov-Hausdorff precompact set of metric measure spaces which is contained in a certain set, called a pyramid, is bounded by some metric measure space with respect to the Lipschitz order inside the pyramid. This is proved as a step towards a possible extension of the statement of Gromov, for which we gave a detailed proof in our previous work. Several related results are also obtained.