Let us consider a sequence of measures $(\mu_n)_n$ which converges weakly to the measure $\mu$ on a metric space $X$. To simplify let us take $X=\mathbb{R}^n$ and $L$ be the Lebesgue measure. Now on $X$ I have a sequence of functions $(f_n)_n$ converging $L$-almost everywhere to a function $f$. This gives a new sequence of measures $$\mu_n(f_n)(A):=\int_A f_n d\mu_n $$

Question: under which conditions the sequence of measures $(\mu_n(f_n))_n$ weakly converge to $\mu(f)$?



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