Simultaneous Strong Law of Large Number classes?

Say that $C$ is a SSLLN class of subsets of some topological space $V$ provided that for every sequence of i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\frac1n \sum_{k=1}^n 1_A(X_k) \to P(X_1\in A)$.

(The important thing is the order of quantifiers. The Strong Law of course implies that for every Borel measurable $A$, we have $\frac1n \sum_{k=1}^n 1_A(X_k) \to P(X_1\in A)$ except on some null set, but the point of being an SSLLN class is that a single exceptional null set that works for all the members of $C$.)

Is there an accepted name for SSLLN classes and any work on conditions for a class of sets to be an SSLLN class?

The only relevant thing I've been able to find in the literature are Glivenko-Cantelli classes. Obviously, any universal Glivenko-Cantelli classes is an SSLLN class. But being an SSLLN class should be much easier, since in the universal GC classes, there is a.s. uniform convergence, while I just want uniform convergence. And of course any countable union of SSLLN classes is an SSLLN class.

I suppose it would be too much to hope for if, for nice spaces $V$, every SSLLN class was a countable union of GC classes.

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