Consider the topological space $X=[0,1]^{[0,1]}$ of functions from $[0,1]$ to itself with the product topology. It is compact by Tychonoff's theorem. Consider a sequence of measurable functions $f_n\in X$ indexed by $n\in\mathbb N$. By compactness, there exists a converging subnet of $f_\bullet$. Is its limit necessarily measurable? If yes, what is an argument for it, if no, what is a counterexample?
One the one hand, there are nets of measurable functions that do not have any measurable accumulation point. On the other hand, the dominated convergence theorem asserts that limits of subsequences are measurable. The question at hand sits in between: it is concerned with sequences, not nets, and subnets, not plainly subsequences.
