Weak topology and weak convergenge in probability spaces

Question

Let $X$ be a Polish space (metrizable, complete, separable) with $\mathcal{B}(X)$ its borel sigma algebra.

Let us consider $\mathcal{P}(X)$ the space of probability measures on $\mathcal{B}(X)$. We endow it with the weak topology denoted by $\tau^{w}$ given by the functionals \begin{align} Q\to L_{f}(Q) := \int_{X} f(x) Q(dx),\hspace{0.3cm}f\in C_{b}(X),Q\in \mathcal{P}(X) \end{align} with $C_{b}(X)$ the continuous and bounded real functions, that is a base for the weak topology is given by the sets \begin{align} W(f,x,\delta) : = \{ \mu\in \mathcal{P}(X),\hspace{0.3cm} |L_f(\mu) - x|<\delta \}. \end{align}

This topology induces a convergence, and it can be proved that this convergence is equal to the one given by the functionals, that is we say that a sequence $(\mu_n)_{n\in\mathbb{N}}$ in $\mathcal{P}(X)$ converges weakly to $\mu\in\mathcal{P}(X)$, and we write $\mu_n\rightharpoonup\mu$ if \begin{align} L_f(\mu_n)=\int_Xf d\mu_n \to \int_Xfd\mu = L_f(\mu),\hspace{0.3cm}\forall f\in C_b(X). \end{align}

Now I want to do what it is written here with the 2-Wasserstein distance, that is to check the mesurability of a markov kernel compounded with a Wasserstein distance, and I want to understand the details.

1. First of all, we are interested in the borel sigma algebra of $(\mathcal{P}(X),\tau^w )$, let us denote such sigma algebra as $\mathcal{B}(\mathcal{P}(X))$. Since it can be proved that $(\mathcal{P}(X),\tau^w )$ is Polish as it is stated in this other question, it has a countable base, so it follow that $\mathcal{B}(\mathcal{P}(X))$ is equal to the smallest sigma algebra that makes all $(L_f)_{f\in C_b(X)}$ measurable, and so we can check the measurability of a Markov Kernel by compounding it with $L_f$ for all $f\in C_b(X)$, is this right?

2. Now I asked myself something more general. For example in the book of Villani, Optimal transport, old and new (2008), chapter 6, it is stated that \begin{align} \mathcal{P}_p(X) = \{ \mu \in \mathcal{P}(X): \int_X d(x_0,x)^p \mu(dx)<+\infty \} \end{align} is a Polish space with the $p-Wasserstein$ $distance$, and he proves that \begin{align} \mu_n \rightharpoonup \mu, \hspace{0.3cm}and\hspace{0.3 cm} \int_X d(x,x_0)\mu_n \to \int_X d(x,x_0) \mu \iff W_p(\mu_n,\mu) \to 0. \end{align} Now I was wondering, he defines another notion of convergence and he proves that the Wasserstein distance metrize such convergence. But what about the weak topology induced by the functional that define such convergence, that is \begin{align} \{L_f: f\in C_b(X), x \to d(x,x_0)^p, x_0\in X \}? \end{align} In general if a distance metrize the weak convergence it is not true that it induces the same topoly, I think about $l^1$ where the strong topology and the weak topology induce the same sequences that converge (as said by Brezis), but the strong and the weak topology are different. In this case we have a distance that induces the same sequences which converge, but the topology a priori may be different. When Villani talks about continuity and $lsc$ about $W_p$, in chapter 6, he always intend sequential continuity and sequential semicontinuity right? He never speak about the weak topology or am I wrong? When he talks about the fact that $\mathcal{P}_p(X)$ is Polish, he intends with respect to the topology induced by the Wasserstein right? Not the weak topology induced by the functionals? However, we are interested in mesurability, so if we see just the sigma algebra generated by the $W_p$ and the borel sigma algebra on $\mathcal{P}_p(X)$, they are the same I think, and it can be proved by using exactly the $lsc$ with respect to the weak topology on $\mathcal{P}(X)$ and the fact that such topology is metrizable and separable, as I said above. Does it follow in some easy way that the borel sigma algebra of the functionals is the same as the borel sigma algebra of the Wasserstein metric? We just need to prove that $W_p$ metrizes the weak convergence and we have the equality between sigma algebras?

3. More generally, let us consider a set $Y$ with some functional $(L_f,f\in \mathcal{F})$ that induces the weak topology $\tau^w$ on $Y$,and it induces a weak convergence on $Y$ given by \begin{align} y_n \rightharpoonup y \iff L_f(y_n)\to L_f(y), \hspace{0.3cm} \forall f\in \mathcal{F}. \end{align} Let us suppose that we succed to find a distance $d$ that metrize the weak convergence, that is \begin{align} y_n \rightharpoonup y \iff d(y_n,y) \to 0. \end{align} When it is true that the borel sets of the weak convergence, the borel sets of the distance $d$, and the smallest sigma algebra that makes $(L_f,f\in\mathcal{F})$ measurable are equal? That is we have \begin{align} \mathcal{B}(\tau^w) = \mathcal{B}(d) = \sigma( L_f,f\in\mathcal{F} ) ? \end{align} I think that we may need that $d$ induces a metric space searable, but what else?

Answer 1

• On the space of probability spaces $\mathcal{P}(X)$ on a Polish space $X$, the topology of weak convergence is metrizable. The Wasserstein metrics provide such metrications, in fact they are complete and make the space separable.

• In general, the weak convergence topology on finite Radon measures $\mathcal{M}(X)$ on a Hausdorff space $X$ is defined as the weak topology $\sigma(\mathcal{M}(X),\mathcal{C}_b(X))$ and makes the dual of $\mathcal{M}(X)$ to be $\mathcal{C}_b(X)$. The space $\mathcal{P}(X)$ is a closed convex subset of $\mathcal{M}(X)$.

• When $X$ is locally compact Hausdorff, there is another (more natural) weak topology which is related to the Riesz representation theorem. It is known that the dual space of $\mathcal{C}_0(X)$ ( continuous functions vanishing at infinity) is the space $\mathcal{M}(X)$ of finite Radon measures on $X$. The weak-* topology $\sigma(\mathcal{M}(X),\mathcal{C}_0(X))$ makes $\mathcal{M}(X)$ locally convex linear space. When to the set of Radon probability measures $\mathcal{P}(X)$ we obtain a closed convex compact subspace.
  Another topology that is commonly used in this setting, called the vague topology, I obtained by restricting the test functions to $\mathcal{C}_{00}(X)$ (continuous functions with compact support).

• Vague convergence ($C_{00}(X)$ tes functions) and weak convergence $\mathcal{C}_b(X)$ on locally compact metric spaces are related by the following result

On a locally compact metric space $S$, a net of nonnegative Borel measures (probabilities in particular) $ \mu_\alpha$ converges weakly to another nonnegative Borel measure $\mu$ iff $\mu_\alpha$ converges vaguely to $\mu$ and $\mu_\alpha(S)$ converges to $\mu(S)$.

Volume 2, of Bogachev's Measure Theory, Springer 2007, has a general presentation of weak convergence of measures on topological spaces. It covers also the classical results of Prokhorov for Polish spaces.

O. Hernandez' book on Markov Chains and Invariant probabilities. Springer 2003, has also a good summary related weak convergence.