P_{n} g = n^{- 1} \sum_{i = 1}^{n} g (X_{i})

and

P g = E g (X)

, the class

G

sup_{g \in G} ∣ P_{n} g - P g ∣ ⟶ 0

Fix

ε > 0

and choose finitely many brackets

[g_{j}^{L}, g_{j}^{U}]

L^{1} (P)

width at most

ε

. The weak law of large numbers, simultaneously for their finitely many endpoints, gives

max_{j} {∣ P_{n} g_{j}^{L} - P g_{j}^{L} ∣, ∣ P_{n} g_{j}^{U} - P g_{j}^{U} ∣} \to 0.

g_{j}^{L} \leq g \leq g_{j}^{U}

, bracketing both

P_{n} g

and

P g

shows

∣ P_{n} g - P g ∣ \leq max {∣ P_{n} g_{j}^{L} - P g_{j}^{L} ∣, ∣ P_{n} g_{j}^{U} - P g_{j}^{U} ∣} + P (g_{j}^{U} - g_{j}^{L}) .

Taking the supremum gives a limit superior at most

ε

. Since

ε

is arbitrary, the ULLN follows.

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