# Convergence of probability measure and the *-weak convergence ?

Given a Polish space X, I note $C_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and and $(C_b(X))^\star$ its topological dual with the $*-$weak convergence $\sigma((C_b(X))^\star, C_b(X))$. To a Borel probability $P$ on $X$ we can associate a $l(P) \in (C_b(X))^\star$ by $$E_P[\phi]=< l(P) , \phi >$$for every $\phi \in C_b(X)$, where $<,>$ denotes the duality bracket. I further consider a sequence $(P_n)_{n\in \mathbb{N}}$ of Borel probabilities on $X$ which is such that $(l(P_n))$ converges in the $*-$weak topology of $(C_b(X))^\star$ to a $m \in (C_b(X))^\star$. Obviously $$< m,G_1 >=1$$ (where $G_1$ is the function $G_1(x)=1$ for any $x\in X$), and for any positive $\phi$ $$< m,\phi >\geq 0$$ (the positive cone is closed). However, is it true that one can find a unique probability $P^\star$ on $X$ such that $$m=l(P^\star)$$ ???? The problem could come from the sigma additivity.

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I believe that the answer is affirmative. Weak$^{\ast}$ topology on probability measures can be metrized (e.g. via Lévy–Prokhorov metric). When we deal with a Polish space $X$, then this metric is complete. Using this, I would like to conclude that the set of probability measures is sequentially closed; indeed, weak$^{\ast}$ convergent sequence of probability measures satisfies Cauchy condition, so it has to converge to some probability measure. Nevertheless, this space is definitely not weak$^{\ast}$ closed, as the example of $(\delta_{n}) \subset \ell_{\infty}^{\ast}$ shows.

I hope that it is correct.

EDIT The above argument is completely wrong: of course completely metrizable subset isn't necessarily closed, e.g. $(0,1) \subset \mathbb{R}$. Nevertheless, I found a paper by Dimitris Gatzouras, On weak convergence of probability measures in metric spaces, in which he claims that the set of separably supported Borel probability measures is sequentially closed. I haven't read it, so I can't tell if his argument is correct.

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It is a standard result of descriptive topology that the space of Borel probability measures on a polish space $S$ is itself a polish space with the weak star topology as the dual of $C^b(S)$ (Nota bene, not as a Banach space but with the so-called strict topology, i.e., the finest locally convex topology which agrees with compact convergence on the unit ball). This result is to be found, e.g. in the text on descriptive topology by Kechris.

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Yes, the problem is with sigma-additivity. If the space $X$ is not compact, then the dual of $C_b(X)$ consists of finitely additive measures. Think about the case when your $X$ is just a discrete countable space, so that $C_b(X)\cong \ell^\infty$.
Yes, but i was wondering whether the fact that $m$ was the limit of elements $(l(P_n))$ associated to probabilities (and not any normed positive element of $(C_b(X))^\star$) implies that $P$ was also a probability. Naively i hope that the convergence of the sequence $(l(P_n))$ in the $\star-$weak topology could imply the tightness of $(P_n)$. Do you know whether this last fact is wrong ? –  Theluze Mar 17 at 13:48
"Sigma-additive probability measures are dense in finitely additive probability measures with respect to the weak topology" Do you mean with respect to the $\star-$weak topology ? Is it trivial ? –  Theluze Mar 19 at 11:06