Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an analogue to the Fréchet-Kolmogorov compactness Theorem that provides a tractable way to check if $\mathcal{F}$ is a relatively compact subset of $L^2(\mathcal{M}, P)$?
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$\begingroup$ If I'm not mistaken on a compact manifold $L^p$ convergence is equivalent to local $L^p$ convergence in finitely many charts. And since the Fréchet-Kolmogorov theorem is essentially local (the standard proof clearly shows this!) the usual characterization works just fine, replace "translations" by "small translations in charts". No? $\endgroup$– leo monsaingeonCommented Nov 26, 2023 at 4:54
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The result you mention uses the algebraic structure of euclidean space since it involves a form of uniform approxability of the set and its translates. However, there are many criteria for compactness which dispense with this, e.g., one which involves approximability by conditional expectations rather than translates (see, e.g., p. 295 in Bogachev, Measure Theory, vol. I) and so do not require an algebraic structure on the underlying measure space. There is, in fact, a considerable literature on compactness in $L^p$-spaces which goes back to the 30's. You might want to have a look at a recent article by Krotov in Sb. Math. 203:7 (2012) 1045--1064 which gives a brief historical summary in the introduction.
