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Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
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Function is almost everywhere 1 w.r.t. sequence of regular Borel probability measures
The statement you want to prove seems wrong: for instance on $K=[0,1]$ with $U=]0,1[$ let $(q_n)$ be countable dense in $U$, and let $\mu_n$ be the Dirac measure on $q_n$. Because the measures are Dir …