What follows is a total boundness criterion in the space $L^1(X)$, where $X$ is arbitrary space with probabilistic continuous measure (Lebesgue space). Of course, all such spaces $X$ and hence $L^1(X)$ too are isomorphic, but common criteria of (pre-)compactness use additional structure on $X$ (say, Kolmogorov-Riesz criterion deals with Lebesgue measure on $\mathbb{R}^n$ and uniform continuity of small shifts).
The set $A\subset L^1(X)$ is totally bounded if and only if
(i) for any $\varepsilon>0$ there exists $\delta>0$ such that $\int_Y |f(x)|<\varepsilon$ provided that measure of $Y\subset X$ is less then $\delta$ (uniform integrability).
(ii) (universal partition) for any $\varepsilon>0$ there exist a finite partition $X=\sqcup_{i=1}^n X_i$ with the following property: for any $f\in A$ there exists an ("exceptional") subset $Y\subset X$ of measure at most $\varepsilon$ such that $|f(x)-f(y)|\leq \varepsilon$ for all $i$ and all $x,y\in X_i\setminus Y$.
Kolmogorov-Riesz criterion less or more corresponds to partition of $\mathbb{R}^d$ onto small cubes.
The question is where I may find a reference to such or similar criterion.
