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Bochner's theorem says that if $M$ is a compact Riemannian manifold withnegative Ricci curvature, then the isometry group $\operatorname{Iso}(M)$ isfinite. In this article, we show that if $(X,d,m)$ is a compact metric measurespace with synthetic negative Ricci curvature in Sturm's sense, then themeasure preserving isometry group $\operatorname{Iso}(X,d,m)$ is finite. Wealso give an effective estimate on the order of the measure preserving isometrygroup for a compact weighted Riemannian manifold with negative Bakry-\'EmeryRicci curvature except for small portions.

The Measure Preserving Isometry Groups of Metric Measure Spaces