# David Fremlin: On Prokhorov spaces

Place: Fields Institute (Room 210)

Date: September 18th, 2015 (13:30-15:00)

Speaker: David Fremlin

Title: On Prokhorov spaces

Abstract:  If $X$  is a Hausdorff space, we have a narrow’ topology on the set $PX$ of Radon probability measures on $X$ generated by sets of the form $\{\mu:\mu G>\alpha\}$  for open subsets $G$ of $X$ and real numbers $\alpha$.  It can be tricky to understand which sets of measures are compact for this topology.  A set $A\subseteq PX$ is uniformly tight’ if for every $\epsilon>0$ there is a compact $K\subseteq X$ such that $\mu(X\backslash K)\le\epsilon$ for every $\mu\in A$;  uniformly tight sets are relatively compact.  $X$ is `Prokhorov’ if relatively compact sets are uniformly tight. Cech-complete spaces are Prokhorov;  D.Preiss showed that the space of rational numbers is not Prohorov.  I explore some further cases.