Definition of weak convergence of probability measures and weak-* convergence

Question

In Wikipedia (link), we have the following definition:

$S$ is a metric space, considered with its Borel $\sigma$-algebra. Then, a sequence of Borel probability measures $P_n$ on $S$ converges weakly to a Borel probability measure $P$ on $S$ if for every bounded continuous function $f_n:S\rightarrow\mathbb{R}$, $\int fdP_n\longrightarrow\int fdP$.

It is noted that this is "actually" weak-* convergence.

How is it weak-* convergence? For weak-* topology, I need a normed space, and then I need to take a dual and there I have weak-* convergence. Probably the space of probability measures lives as a subspace of this dual.

I am confused about this, and would appreciate some accurate definitions of weak and weak-*, and a description of the relations between them.

I only care about the Borel probability measures on $S$, and I mostly care about the case where $S$ is a compact metric space.

Answer 1

First suppose $X$ a Banach space and let $X^{\star}$ be it topological dual, i.e, the set of all continuous linear functionals $l:X\to \mathbb{R}.$ The weak topology in $X$ is the smallest topology in which all functionals $l\in X^{\star}$ remains continuous. More precisely the weak topology has the following basis of neighbourhoods: $$ \bigcap_{\text{finite}}l_{i}^{-1}((a_i,b_i)), ~~ (a_i, b_i)\subset \mathbb{R} $$ where $a_i$ or $b_i$ could be $\pm \infty$.

The $weak$-$\star$ topology in $X^{\star}$ is the smallest topology in $X^{\star}$ for which all linear functionals $\varphi_{x}\in X^{\star \star}, x\in X$, of the form $\varphi_{x}(l)=l(x)$, are continuous. More precisely the weak topology has the following basis of neighbourhoods: $$ \bigcap_{\text{finite}}\varphi_{x_i}^{-1}((a_i,b_i)), ~~ (a_i, b_i)\subset \mathbb{R}. $$ where $a_i$ or $b_i$ could be $\pm \infty$

It is a straightforward calculation to see that $(l_n)\subset X^{\star}$ converges to $l$ in the $weak$-$\star$ topology iff $\varphi_{x_i}(l_n)\to \varphi_{x_i}(l)$ for all $x_i$.

Now consider $X=C_b(S)$, then $X^{\star}=rba(S)$ (regular Borel additive measures.) Then if we consider in $X^{\star}$ the $weak$-$\star$ topology we have by the above exposed that $(\mu_n)\subset X^{\star}$ converges to $\mu \in X^{\star}$ iff we have for all $f\in C_b(S)$ that $$ \varphi_f(\mu_n)=\int f d\mu_n=\int f d\mu =\varphi_{f}(\mu). $$