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It is well known that if $X$ is a Polish space and $\mathcal{F} \subset \mathcal{M}_+(X)$ (the set of finite positive Radon measures on $X$) is uniformly tight and bounded in mass, it is relatively compact w.r.t. to the weak topology, i.e. the coarsest topology on $\mathcal{M}_+(X)$ w.r.t. the maps $\mu \mapsto \int_X \varphi \text{ d} \mu$ are continuous for every $\varphi \in C_b(X)$, the continuous bounded real functions on $X$.

I am intersted in a similar statement but in the general case of a Hausdorff topological space $X$.

A possible way to introduce a topology on $\mathcal{M}_+(X)$, when $X$ is a Hausdorff topological space, is to consider the coarsest topology on $\mathcal{M}_+(X)$ w.r.t. the maps $\mu \mapsto \int_X \varphi \text{ d} \mu$ are lower semi continuous for every $\varphi \in LSC_b(X)$, the lower semi continuous and bounded real functions on $X$.

In the book of Schwarz "Radon measures on arbitrary topological spcaes" it is proven that, in this topology, uniform tightness and boundedness in mass together again imply relative compactness.

I am wondering, how much can I enrich the topology on $\mathcal{M}_+(X)$ and still have that Prokhorov theorem holds?

For example, if $X$ is a Hausdorff topological space and I endow $\mathcal{M}_+(X)$ with the coarsest topology w.r.t the maps $\mu \mapsto \int_X\varphi \text{ d}\mu$ are continuous for every $\varphi \in LSC_b(X) \cup USC_b(X)$, what happens?

Here $LSC_b(X)$ (resp. $USC_b(X)$) is the set of the lower (resp. upper) semi continuous bounded real functions on $X$.

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  • $\begingroup$ I have problems with your text. What do you mean by "continuous on varying of $\phi$? The last sentence is dubious. Why should $\mu \o \int \phi d\mu$ be continuous for semicontinuous $\phi$? Do you know the notes on p. 454 and 455 in Bogachev, Measure Theory 2, Springer 2007? $\endgroup$ – Dieter Kadelka May 4 '20 at 8:39
  • $\begingroup$ I have edited the text, I hope it is clear now. I am just considering different initial topologies on $\mathcal{M}_+(X)$. I've just read those notes: I think I will surely find something useful in the papers suggested there! $\endgroup$ – Bremen000 May 4 '20 at 8:48
  • $\begingroup$ Much clearer now! But I still have a problem with your last sentence. Being both u.s.c. and l.s.c. implies continuity of a function. $\endgroup$ – Dieter Kadelka May 4 '20 at 9:05
  • $\begingroup$ I mean for functions that are upper or lower semi continuous. Edited again, now it should be clear! $\endgroup$ – Bremen000 May 4 '20 at 9:59
  • $\begingroup$ Your topology at the end is finer than the weak topology, and may be much much finer. For instance, take $X = [0,1]$; since the functions $1_{\{x\}}$ are USC, I think the set of Dirac masses is discrete in this topology, and certainly not relatively compact even though uniformly tight. $\endgroup$ – Nate Eldredge May 4 '20 at 12:29
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This seems to be a pure general topology problem. Enriching the topology will necessarily destroy the Prohorov property whenever the enrichment matters. If you have two nested Hausdorff topologies and a set is relatively compact in the finer topology, then the closure is the same under both topologies and the trace topologies coincide on the closure.

To see this, let $\tau$ and $\tau'$ be Hausdorff topologies on $X$ such that $\tau\subseteq\tau'$ and let $R\subseteq X$ be relatively compact in the topology $\tau'$. Trivially, $R$ is also relatively compact in the coarser topology $\tau$ and the $\tau'$-closure of $R$ is a subset of the $\tau$-closure of $R$. Let $x$ be a point in the $\tau$-closure of $R$ and $\langle x_\alpha\rangle$ be a net in $R$ converging to $x$ under $\tau$. Since $R$ is relatively compact under $\tau'$, a subnet will $\tau'$ converge to a point $x'$. But this subnet will also converge to $x'$ under $\tau$. Since $\tau$ is Hausdorff , $x=x'$ and the closure of $R$ is the same under both topologies. Let $C$ be this compact closure. The identity is a continuous function from $(C,\tau')$ to $(C,\tau)$ and will therefore map compact subsets of $C$ to compact subsets of $C$. But these are exactly the closed subsets of $C$ under both topologies, so both topologies must coincide on $C$.

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