= Solution
Writing $P_ng=n^{-1}\sum_{i=1}^ng(X_i)$ and $Pg=\mathbb Eg(X)$, the class $\mathcal G$ satisfies a <uniform law of large numbers> when
$$
\sup_{g\in\mathcal G}|P_ng-Pg|\longrightarrow0
$$
in probability; a strong ULLN uses almost-sure convergence.
Fix $\varepsilon>0$ and choose finitely many brackets $[g_j^L,g_j^U]$ of $L^1(P)$ width at most $\varepsilon$. The <weak law of large numbers>, simultaneously for their finitely many endpoints, gives
$$
\max_j\{|P_ng_j^L-Pg_j^L|,|P_ng_j^U-Pg_j^U|\}\to0.
$$
If $g_j^L\leq g\leq g_j^U$, bracketing both $P_ng$ and $Pg$ shows
$$
|P_ng-Pg|\leq\max\{|P_ng_j^L-Pg_j^L|,
|P_ng_j^U-Pg_j^U|\}+P(g_j^U-g_j^L).
$$
Taking the supremum gives a limit superior at most $\varepsilon$. Since $\varepsilon$ is arbitrary, the ULLN follows.
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