# Almost sure convergence of the supremum over a class of random variables

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?