 Uniform strong law from finite L1 bracketing

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 ${\sup }_h|P_{n}h-Ph|\to 0$ on a common probability-one event. For one finite $\epsilon$-cover, the supremum is bounded by $\epsilon$ 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.