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Assume $\mu_n$ is a family of probability measures and $f_m\to 0$ with $|f_m(x)|\le V(x)$. If we have $\sup_{n\ge 1}\int_{E} V(x)^2\mu_n(dx)<\infty$.

Prove that $\lim\limits_{m\to\infty}\sup\limits_{n\ge 1}\int_{E} f_m(x) \mu_n(dx)=0$.

I know that this might be in realtion with uniformly integrablility, but I can not prove it directly. Can you help me?

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  • $\begingroup$ Where does this come from? There are very easy counter-examples (assuming that $f_m\to 0$ is supposed to be point-wise) $\endgroup$ Commented Nov 20 at 6:36

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