Almost sure convergence of a net Let $X$ and $X_\varepsilon$, $\varepsilon > 0$ be random variables on $\mathbb{R}$ (or any metric space) defined on the same probability space $(\Omega, \mathcal{A}, P)$. Assume that $X_{\varepsilon_n}$ converges to $X$ $P$-almost surely for all sequences $\varepsilon_n \to 0$. Does it follow that $X_\varepsilon$ converges to $X$ $P$-almost surely as $\varepsilon \to 0$?
For real functions on metric spaces continuity is equivalent to sequential continuity. But the issue here is that we have a probability measure in between. Also, the set of all null sequences is uncountable, so we can get measurability issues.
 A: No. The trouble is that one can find an uncountable family of disjoint sequences of positive real numbers converging to $0$. If this family is $(\epsilon_n^i)_n, i\in I$ (where $I$ is some uncountable set), then the numbers $\epsilon_n^i$ are all distinct from one another, so the variables $X_{\epsilon_n^i}$ can be defined any way we like, independently of one another. Thus, for fixed $i$, we can make the sequence $X_{\epsilon_n^i}$ fail to converge to $X$ on some negligible set $Z_i$, but such that $\bigcup_{i\in I} Z_i$ is non-negligible. Therefore $X_\epsilon$ will not converge a.e. to $X$ as $\epsilon\to 0$. And we can get sequential convergence of course, as long as we define $X_\epsilon$ correctly for all the other values of $\epsilon$, say $X_\epsilon=X$.
Let's just take $X=0$, and the underlying probability space to be $[0, 1]$. We need a few building blocks to make this construction work:


*

*An uncountable family $Z_i$ of negligible sets whose union is non-negligible. Easy, take $Z_i=\{i\}$ for $i\in[0,1]$.

*Given a negligible set $Z$, a sequence of random variables which converges to $0$ everywhere but on $Z$. Just take $X_n=0$ off of $Z$ and $X_n=1$ on $Z$.

*An uncountable family of disjoint sequences $\epsilon_n^i$ converging to $0$. We can take $\epsilon_n^i=i\frac1n + (1-i)\frac1{n-1}$ for $i\in[0, 1)$, say.
Just for fun, we can write down an explicit formula for the variable $X_x$, $x\in[0, 1)$, all defined on the probability space $[0, 1)$. We would have $X_x(y)=1$ for $y=\frac{x-\frac1{\lceil x\rceil-1}}{\frac1{\lceil x\rceil}-\frac1{\lceil x\rceil-1}}$ and $0$ otherwise.
