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Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume for each $n$, $Y_n:\Omega\rightarrow\mathbb{R}$ is a function but $Y_n$ is not necessarily $\mathscr{F}$-measurable. In this case, is it still meaningful to talk about almost sure convergence of $Y_n$? Conceptually, we can define almost sure convergence as $$\exists\hat{\Omega}\in\mathscr{F}\quad\mathrm{such\,\,that}\quad P(\hat{\Omega})=1\quad\mathrm{and}\quad \{Y_n(\omega)\}\,\,\mathrm{converges}\,\,\forall \omega\in\hat{\Omega}.$$

In every probability textbook I have, they all define almost sure convergence for "random variables". But I think what I mentioned might arise naturally in some situations. For example, if for each $n$, $\{X^n_\lambda\}_{\lambda\in \Lambda}$ is a class of random variables where $\Lambda$ is uncountable, then $$Y_n\equiv\sup_{\lambda}X_\lambda^n$$ is not necessarily measurable, but still we sometimes want to talk about convergence property of $\{Y_n\}$.

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    $\begingroup$ You may as well ask what you can do with non-measurable random variables $X$. You can't integrate them, nor find a CDF, or even ask what the probability of say, $X=1$ is. Ultimately, you can make anything you want converge to something if you define it properly but the usefulness of it leaves something to be answered. $\endgroup$ – Alex R. May 1 '15 at 23:44
  • $\begingroup$ "But I think what I mentioned might arise naturally in some situations." What I think is that you would have to be much more specific before this conviction of yours becomes convincing. For example, in which situations would one "want to talk about (pointwise or almost sure) convergence properties of" the random sequence $(Y_n)$? $\endgroup$ – Did May 2 '15 at 10:43
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Not only a.s. convergence but pointwise convergence, as well, can be defined in the case of sequences of non measurable functions. Let, for instance, $$([0,1],\mathscr A=\left\{\emptyset,[0,1/2],(1/2,1],[0,1]\right\},\mathbb P((0,1/2]))=\mathbb P((1/2,1])=1/2)$$ be a probability space, and let $$X_n(\omega)=\frac{\omega}{n}, \text{ if }\ \omega\in[0,1].$$ Obviously $X_n$ converges pointwise to $0$ on $[0,1]$. So far so good. However, there is no answer to important$^*$ questions. Consider only the following example: $$\mathbb P\left(X_3<\frac{1}{5}\right)=\ "\mathbb P"\left(\left\{\omega:0\le \omega<\frac{3}{5}\right\}\right)=??$$ There is no answer because $\mathscr A$ and $\mathbb P$ could be extended many different ways.

$^*$ Philosophcal-BTW: What is important at all?

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