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Let $(X, d)$ be a metric space. Then @Nate Eldredge said that

The space $\mathcal{P}(X)$ of probability measures on a Polish space $X$, endowed with the weak topology (induced by bounded continuous functions), is first countable - indeed Polish. This is a standard result that you can find in, say, Billingsley's Convergence of Probability Measures. Note here that the "weak topology" is really a weak$^*$ topology. You can think about the fact that, given a separable Banach space $W$, the weak* topology on $W^*$ is not first countable, but the topology it induces on the ball of $W^*$ is first countable (indeed, Polish!).

  1. In Onno van Gaans's note Probability measures on metric spaces, I got that
  • Theorem 4.2 If $X$ is separable, then the weak topology of $\mathcal{P}(X)$ is separable.
  • Proposition 5.3 If $X$ is compact, then the weak topology of $\mathcal{P}(X)$ is compact.

Could you elaborate on which theorem in Billingsley's Convergence of Probability Measures mentiones that if $X$ is Polish then the weak topology of $\mathcal{P}(X)$ is Polish?

  1. From Riesz–Markov–Kakutani theorem, it seems to me weak$^*$ topology is induced by the space $C_0 (X)$ of continuous functions on $X$ vanishing at infinity, but I'm not sure. Could you elaborate on how the weak$^*$ topology of $\mathcal{P}(X)$ is defined?
Analyst
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  • Your broad question ("is $\mathcal{P}(X)$ polish when $X$ is?") has been asked before, see here, for instance. I'm not voting to close because it seems like you might want a reference in Billingsley in particular? Or maybe you want more information on what the weak topology on $\mathcal{P}(X)$ is? Either way, you can find this as theorem 17.23 in Kechris's Classical Descriptive Set Theory. This is in chapter 17.E, which is all about the space of (borel) probability measures on a polish space $X$. – HallaSurvivor Oct 21 '22 at 00:55
  • @HallaSurvivor Thank you for answering my first question! Do you have any hint for the second one? – Analyst Oct 21 '22 at 06:37
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    The weak-* topology of $\mathcal{P}(X)$ is, as I said, the one induced by the bounded continuous functions. Specifically, the weakest topology such that for every bounded continuous function $f$, the map $\mu \mapsto \int f,d\mu$ is continuous. If $X$ is locally compact, then this is the same as the topology induced by $C_0(X)$, but in general it is not. – Nate Eldredge Oct 21 '22 at 07:12
  • @NateEldredge The term "weak$^$ topology" usually arises from the weak$^$ topology of the dual space of a TVS. If we use the term "weak$^$ topology" for the one induced by $C_0(X)$, then it makes sense by Riesz–Markov–Kakutani theorem. However, I could not understand how we use the term "weak$^$ topology" for the one induced by $C_b(X)$. Could you elaborate more? – Analyst Oct 21 '22 at 07:38

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