= L1 Kolmogorov-Riesz compactness criterion
{c}
{title2=$S\Subset L^1(\mathbb R)\iff S\text{ bounded, uniformly tight and translation continuous}$}
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 $\sup_{f\in S}\|f(\cdot+h)-f\|_1\to0$ as $h\to0$. 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>.
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