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Good night, we have the following:

Let $(X,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous functions on $Y$, equipped with the metric $||f||=\sup_Y|f|$. Then a sequence $\{\mu_n\}\in\mathcal{M}(Y)$ converges weak star to $\mu\in\mathcal{M}(Y)$ if $\mu_n(f)\longrightarrow \mu(f)$ for all $f\in C_B(Y)$.

I would like to know if a sequence $\{\mu_n\}\in\mathcal{M}(Y)$ converges weak star to $\mu\in\mathcal{M}(Y)$ then $\mu_n(f)\longrightarrow \mu(f)$ is satisfied for a larger class of functions, for example functions in $L^1$.

I thank any help if as a reference.

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    $\begingroup$ It's not a research question. If $f$ is any discontinuous function then there exists a sequence $(x_n)$ in $X$ such that $x_n \to x \in X$ but $f(x_n) \not\to f(x)$. Letting $\mu_{x_n}$ be the Dirac measure at $x_n$, we then have $\mu_{x_n} \to \mu_x$ weak* (although I am not sure this term is appropriate) but $\int f\, d\mu_{x_n} \not\to \int f\, d\mu_x$. $\endgroup$
    – Nik Weaver
    Commented May 25, 2016 at 3:56
  • $\begingroup$ You start your post with $(X,d)$, but $X$ is never mentioned again in your post, only $Y$. Typo, perhaps? $\endgroup$
    – Martin Sleziak
    Commented May 25, 2016 at 5:01
  • $\begingroup$ If, misspelling, I meant $(Y,d)$. $\endgroup$
    – Rusbert
    Commented May 25, 2016 at 19:21

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