Vague topology and dual space

Question

I am having a problem in measure convergence and functional analysis. I am not familiar with this fields so any further reference is welcome.

Suppose that $X = (a,b) \times \mathbb{T}^d$ where $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$ is the $d$-dimensional torus. Let $M(X)$ be the spaces of signed Radon measures with vague topology:

$$ \mu_n \to \mu, \text{ if and only if } \int_X f d\mu_n \to \int_X f d\mu \text{ for all compactly supported } f $$

I have a sequence $\{\mu_n\}$ s.t. $\sup_{n} \left\vert \mu_n \right\vert (X) < \infty$. Can I apply Banach-Alaoglu theorem to conclude that there is a vague convergent subsequence of $\{\mu_n\}$?

It seems to me that my question is essentially the followings:

Is $M(X)$ the dual space of some spaces of functions e.g. $M(X) = (C_c(X))^*$ or $M(X) = (C_b(X))^*$?
Does vague (weak$^*$) compactness imply sequentially vague (weak$^*$) compactness in my case?

I think you need to assume that each $\mu_n$ is also a Radon measure. — Evangelopoulos Foivos, Sep 22 '22 at 14:05
If I recall correctly, $M(X) = C_c(X)^* = C_0(X)^$ when $X$ is locally compact. Now, $C_0(X)$ is separable whenever $X$ is locally compact and second countable and so the closed unit ball of $C_0(X)^$ is metrizable with the weak-star (vague) topology. This will allow you to extract a vaguely convergent subsequence. — Evangelopoulos Foivos, Sep 22 '22 at 14:24
@EvangelopoulosF. Thank you for the reply! By $C_0(X)$, do you mean the space of functions that vanish at infinity? Or you simply mean the space of bounded continuous functions? If it is the former, then what's the meaning of "at infinity"? In the context $(a,b)$ means a bounded open interval. — Justin Lien, Sep 22 '22 at 16:06
See this https://math.stackexchange.com/questions/266677/equivalent-definitions-of-c-0x — Evangelopoulos Foivos, Sep 22 '22 at 16:54

Vague topology and dual space

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