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Let $\mu_n$ be a sequence of positive radon measures on $\mathbb{R}^n$ weakly converging (as dual of continuous compactly supported functions) to a measure $\mu$.

Assume that $f_n(z)$ is a sequence of positive, compacly supported functions such that: they are uniformily supported in a ball, i.e.: for every $n$ their support is contained in a ball $B_r(0)$ and they are uniformily bounded with $\lvert f_n(z)\rvert\leq C$. Moreover $f_n(z)\to 0$ for any $z\in \mathbb{R}^n$.

Is it true that at least for a subsequence $n_j$, that we have:

\begin{equation} \lim_{j\to\infty}\int f_{n_j}(z) d\mu_{n_j}(z)=0? \end{equation}

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  • $\begingroup$ I have the proof of this fact. I will wait a couple of days if someone comes up with a proof, since I would like to see other proofs. However in a couple of days I will post here the answer. Hints: Cavalieri's formula, Fatou's lemma, and ascoli arzela if you want to prove the statement for the subsequence. It is true in general, though. $\endgroup$ – Diesirae92 Aug 29 '17 at 21:15

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