Convergence of metric transformed spaces

Kazukawa, Daisuke

doi:10.48550/arxiv.2008.04038

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2021

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“…Lemma 24 ( [K,Lemma 4.6]). If X and Y are mm-spaces with Y ≺ ε X for some ε > 0, then there exists an mm-space Z with Z ≺ X and (Y, Z) ≤ 4ε.…”

Section: Proof We Choose Ptmentioning

confidence: 99%

Boundedness of precompact sets of metric measure spaces

Kazukawa¹,

Yokota²

2021

Preprint

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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.Proposition 4. If a set Y of compact metric spaces is GH-precompact, then there exists a compact metric space X with Y ≺ X for any Y ∈ Y.After some preparations in Section 2, we present a proof of Theorem 1 in Section 3. Our proof is a slight modification of Gromov's sketch in [G]. In Section 4, we prove Proposition 4.In our argument, we do not pay careful attention to distinguish an mm-space and an element of X and consider an element x of a product λ∈Λ X λ of a family {X λ } λ∈Λ of sets as a map x : Λ → λ∈Λ X λ with x(λ) ∈ X λ for any λ ∈ Λ.

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“…Lemma 24 ( [K,Lemma 4.6]). If X and Y are mm-spaces with Y ≺ ε X for some ε > 0, then there exists an mm-space Z with Z ≺ X and (Y, Z) ≤ 4ε.…”

Section: Proof We Choose Ptmentioning

confidence: 99%