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Let $X$ be a locally compact Hausdorff space and let $\mu_n$ be a sequence of bounded variation Radon measures on $X$ such that $\int_X g \;d\mu_n \rightarrow \int_X g \;d\mu$ for each $g \in C_0 (X)$ (ie. $g(x) \rightarrow 0$ as $x \rightarrow \infty$ in the one-point compactification of $X$) and $|\mu_n|(X) \rightarrow |\mu|(X)$. Must it hold that $\int_X f \;d\mu_n \rightarrow \int_X f \;d\mu$ for each bounded continuous function $f$, even if $f$ does not tend to zero at infinity?

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I believe this is true, and I think I have seen it in Billingsley's Convergence of Probability Measures. –  Nate Eldredge Dec 26 '11 at 21:44

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