Let the metric measure space \((X,d,\mu)\) be separable and the Borel measure \(\mu\) be finite on every bounded set.
The authors define the pmG-convergence for that type of spaces in three different way using, among others, the Gromov reconstruction theorem [M. Gromov, Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Basel: Birkhäuser (2007;Zbl 1113.53001)] and a variant ofK.-T. Sturm’s distance [Acta Math. 196, No. 1, 65–131 (2006;Zbl 1105.53035)]). Namely, a sequence of pointed metric measure spaces \((X_n,d_n,\mu_n,\bar{x}_n)\) converges to \((X_{\infty},d_{\infty},\mu_{\infty},\bar{x}_{\infty})\) in pmG-sense if there are a complete separable space \((X,d)\) and isometric embeddings \(\imath_n : X_n \to X\), \(n\in\mathbb{N}\cup\{\infty\}\), such that \[ \int \phi d(\imath_n)_{\sharp}\mu_n \to \int \phi d(\imath)_{\sharp}\mu \] for all continuous, bounded functions \(\phi: X\to \mathbb{R}\) which have bounded support.
Then they prove the following properties of this concept:
1) on doubling spaces pmG-convergence coincides with pmGH-convergence (Propositions 3.30 and 3.33);
2) the curvature condition \(\operatorname{CD}(K,\infty)\) is stable with respect to pmG-convergence (Theorem 4.9);
3) the heat flow, defined as the Wasserstein gradient flow of the relative entropy or as the \(L^2\)-gradient flow of the Cheeger energy, is stable with respect to pmG-convergence (Theorems 5.7 and 6.11).

Reviewer: Bożena Piątek (Gliwice)

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