Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space, $\Pi_n : \Omega\to \mathbb{R}$ be a sequence of real-valued random variabes with $\Pi_n \rightarrow \Pi$ almost everywhere. Furthermore, let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a Borel measurable and bounded function. Under these conditions, does it hold, that $h(\Pi_n) \to h(\Pi)$ almost everywhere?

If not, what are some conditions on $h$ under which this holds, which are weaker than continuity?



E.g. let $\Pi_n(\omega)=\frac1n$ and let $\Pi(\omega)=0$ for every $\omega\in\Omega$.

Then $\Pi_n\stackrel{a.s.}{\to}\Pi$ but if $h$ is e.g. the function prescribed by $x\mapsto 1$ if $x>0$ and $x\mapsto0$ otherwise then we do not have $h(\Pi_n)\stackrel{a.s.}{\to}h(\Pi)$.

If you want it to be true in full generality then continuity is necessary and sufficient (bounded is not necessary).

(By questions like these it often pays off to look at degenerated random variables at first)


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