I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability measures, such that the latter fails to be a perfectly normal topological space.

Almost all results start with $\Omega$ separable metric to get nice topological properties on ($\mathcal P(\Omega)$, $\tau$). The answer to the following question seems to be closely related: Gaussian measures on non-separable spaces .

I suppose an easy example would be $\Omega$ normal but not perfectly normal and $\mathcal P(\Omega)$ the set of Dirac measures on $\Omega$, and $\tau$ being the topology induced from that on $\Omega$.

I guess some restriction on the $\tau$ to be "natural" is needed to make this question interesting.