Let $(X,\tau)$ be a topological space and let $\mathcal P$ denote the set of all probability measures on $\mathscr B$, the Borel $\sigma$-algebra generated by the open subsets of $X$. In what follows, let $\mathrm{BC}(X)$ denote the set of bounded, continuous real-valued functions on $X$.

Define a topology $\upsilon$ on $\mathcal P$ as the one generated by the topological subbasis consisting of sets of the form $$U_{f,y,\varepsilon}\equiv\left\{\mathbb P\in\mathcal P:\left|\int_{x\in X}f(x)\,\mathrm d\mathbb P(x)-y\right|<\varepsilon\right\},\quad\text{where } f\in\mathrm{BC}(X),\,y\in\mathbb R,\,\varepsilon>0.$$ The topology $\upsilon$ is often referred to as the weak-star topology, given the functional-analytic structure with which $\mathcal P$ can be endowed (see also below). It is not difficult to check that a net $(\mathbb P_{\alpha})_{\alpha\in A}$ in $\mathcal P$ (where $A$ is a non-empty directed index set) converges to $\mathbb P\in\mathcal P$ with respect to this topology if and only if the corresponding net $$\left(\int_{x\in X}f(x)\,\mathrm d\mathbb P_{\alpha}(x)\right)_{\alpha\in A}$$ of real numbers converges to $\int_{x\in X}f(x)\,\mathrm d\mathbb P(x)$ with respect to the Euclidean topology on $\mathbb R$ for every fixed $f\in\mathrm{BC}(X)$.

It is quite well-known from functional-analytic probability theory that if $(X,\tau)$ is metrizable and compact, then $(\mathcal P,\upsilon)$ is compact. [And I think, although I am not 100% sure, that the Riesz representation theorem ensures the compactness of $(\mathcal P,\upsilon)$ if $(X,\tau)$ is compact and merely Hausdorff, and not necessarily metrizable.]

What I am trying to do is establish the compactness of $(\mathcal P,\upsilon)$ using only the compactness of the underlying space $(X,\tau)$, using no further topological separation or countability axioms:

Conjecture: If $(X,\tau)$ is compact, then so is $(\mathcal P,\upsilon)$.

Naturally, the applicability of the usual functional-analytic toolkit (such as the Riesz representation theorem) is quite limited without further assumptions. I was thinking of using Alexander’s subbasis theorem as a starting point, but I am actually not even sure that the conjecture is true in its general form.

Any hints, references for proofs (if true), or outlines of counterexamples (if false) would be greatly appreciated.

  • $\begingroup$ If you consider a map $\Phi\colon X\to {\bf R}^{C(X)}$, $\Phi(x)(g)=g(x)$, the space $\mathcal P(X)$ and $\mathcal P(\Phi[X])$ should be very closely related, and possibly identical up to taking a Kolmogorov quotient. If that is true, you get a reduction to the case of a compact Hausdorff space. $\endgroup$ – tomasz Jan 10 '17 at 16:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.