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 $L^2(\mathcal{M}, P)$?



Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds
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 (2010) 129-148 which gives a brief historical summary in the introduction.


