Let $X$ be a measurable space, and let $Y$ be a topological space.
If $X$, $Y$ have the same cardinality, is there (necessarily) a measurable bijection from $X$ to $Y$?
2026-03-26 03:00:23.1774494023
Is there measurable bijection?
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Not necessarily. If the sigma algebra on $X$ is countably generated then the sigma algebra on $Y$ would have to be countably generated for the existence of measurable bijection. [ Countably generated means there is sequence $(A_n)$ such that the given sigma algebra equals $\sigma(A_1,A_2,...)$].
Note that irrespective of the cardinality of $X$ we can always fine countably generated sigma algebra on $X$ so all you need is to take $Y$ to have a non-countably generated sigma algebra.