= Uniform strong law from finite L1 bracketing
{title2=$\sup_h|P_nh-Ph|\xrightarrow{\mathrm{a.s.}}0$}
Suppose an integrable measurable-function class can be covered by finitely many <function brackets> of every positive $L^1(P)$ width. For an independent sample with common law $P$, the <empirical measure> satisfies $\sup_h|P_nh-Ph|\to0$ on a common probability-one event. For one finite $\varepsilon$-cover, the supremum is bounded by $\varepsilon$ plus the largest empirical error among its endpoints. The <strong law of large numbers> makes that finite maximum vanish. Taking a countable sequence of widths decreasing to zero proves the assertion. A measurable supremum can be obtained from a <pointwise separable function class>; otherwise the probability-one-event formulation expresses the same pathwise conclusion.
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