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@Echo For 1., if $X$ is compact, then theorem holds. I would like to ask if it still holds in case $X$ is only locally compact (and possibly separable).
The space $\mathcal M(X)^*$ is quite hard to imagine. Could you provide a map $\varphi \in \mathcal M(X)^*$ such that $\varphi (\mu_n)$ does not converge to $\varphi (\mu)$?
@DieterKadelka Do you mean by $\mathcal C_b(X) \subset \mathcal M(X)^*$ that there is an isometrically isomorphic embedding from $\mathcal C_b(X)$ to $\mathcal M(X)^*$?
@DieterKadelka A function $f \in \mathcal C_b (X)$ vanishing at infinity means that for every $\epsilon>0$ the set $\{x:|f(x)| \geq \epsilon\}$ is compact.
