I try to find a counterexample that if $X_n\xrightarrow{\text{a.s.}}X$ then for a measurable function $g:\mathbb{R}\to \mathbb{R}$ this does not imply that $$g(X_n)\xrightarrow{\text{a.s.}}g(X)$$
Unfortunately I have no idea.
2026-04-13 07:03:08.1776063788
Counterexample that a measurable function does not guarantee almost sure convergence
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This would be true if $g$ were continuous with no other assumptions. If you make $g$ discontinuous then "generically" this convergence fails.
To be more specific, here is a simple approach: assume $X_n$ converges a.s. to some fixed constant $c$, but that the $X_n$ are never equal to $c$. Then define $g(c)$ to have one value and $g(x)$ to have some other value when $x \neq c$.