We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov's book [3]. We obtain a complete answer for this question.
The notion of the metric measure foliation is introduced by Kell, Mondino, and Sosa in [7]. They studied the relation between a metric measure space with a metric measure foliation and its quotient space. They showed that the curvature-dimension condition and the Cheeger energy functional preserve from a such space to its quotient space. Via the metric measure foliation, we investigate the convergence theory for a sequence of metric measure spaces whose dimensions are unbounded.
We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the convergence of the original sequence from that of the transformed sequence, respectively. As an application, we prove that spheres and projective spaces with standard Riemannian distance converge to a Gaussian space and the Hopf quotient of a Gaussian space, respectively, as the dimension diverges to infinity.
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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