Let $\mathcal{X}_n=\{ X_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\Lambda$. Assume that for any $\lambda \in \Lambda$ it is known that $X_{n,\lambda}\to 0$ almost surely, i.e. for any $\epsilon>0$ $$ P\{|X_{n,\lambda}|>\epsilon \text{ i.o. }\}=0. $$ Under which assumptions can we conclude that $P\{\sup_{\lambda \in\Lambda}|X_{n,\lambda}|>\epsilon \text{ i.o. }\}=0$ for all $\epsilon>0$?
I know that this problem is well studied in the case where $Y_1, Y_2, \ldots$ are independent and identically distributed random vectors, with common distribution $Q$, and each $X_{n,\lambda}$ is of the form $$ \frac{1}{n}\sum_{i=1}^nf_\lambda(Y_i)-\int f_\lambda(y)\text{d}Q(y), $$ where $\mathcal{F}=\{f_\lambda, \lambda \in \Lambda\}$ constitutes a class of measurable functions. In this context, appropriate metric entropy conditions yield a functional generalisation of the strong law of large numbers. Are you aware of results which hold true beyond the case of random variables in the form of centered means?
