 Solution

Writing $P_{n}g=n^{-1}{\sum }_{i=1}^{n}g(X_i)$ and $Pg=\mathbb{E}g(X)$, the class $\mathcal{G}$ satisfies a uniform law of large numbers whenin probability; a strong ULLN uses almost-sure convergence.

Fix $\epsilon >0$ and choose finitely many brackets $[g_j^L,g_j^U]$ of $L^1(P)$ width at most $\epsilon$. The weak law of large numbers, simultaneously for their finitely many endpoints, givesIf $g_j^L\le g\le g_j^U$, bracketing both $P_{n}g$ and $Pg$ showsTaking the supremum gives a limit superior at most $\epsilon$. Since $\epsilon$ is arbitrary, the ULLN follows.