L1 Kolmogorov-Riesz compactness criterion ($S⋐L^1(\mathbb{R})⟺S\text{ bounded, uniformly tight and translation continuous}$)

A subset of $L^1(\mathbb{R})$ is relatively norm compact exactly when it is bounded, its absolute-integral tails tend to zero uniformly, and ${\sup }_{f\in S}\Vert f(\cdot +h)-f{\Vert }_1\to 0$ as $h\to 0$. Necessity follows by taking a finite norm net and using the tail decay and translation continuity in Lp of its finitely many centers; the error contributes at most twice the net radius. For sufficiency, convolve with a triangular unit-integral kernel of width $\delta$. Translation continuity gives uniform approximation in $L^1$, while boundedness gives a common supremum bound and a common Lipschitz constant for the smoothed family. The original tail condition controls its tails outside a slightly enlarged interval. On that interval Arzelà-Ascoli theorem gives a finite uniform net. Extend its centers by zero outside the interval to obtain a finite $L^1$ net, and combine the smoothing and tail errors. Completeness of $L^1$ turns total boundedness into relative compactness.