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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