Can we recover a topological space from the collection of Borel probability measures living on it? Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
 A: I write this as an answer due mainly to space constraints.    I want to   elaborate a bit on Nate Eldredge's useful comments. 
Think of   Gelfand's classical  theorem stating that a compact space $X$ is determined by the  space $C(X)$ of continuous complex valued functions   on $X$.  This  statement is more   precise than this.  
The space  $C(X)$  is   given an algebraic-topologic structure  (commutative $C^*$-algebra). This  structure alone completely determines the space $X$ as the spectrum of this algebra equipped with an appropriate topology. Continuous maps between two compact spaces $X_1$ and $X_2$ induce continuous morphisms between the corresponding $C^*$-algebras,  and the isomorphisms of $C^*$-algebras induces homeomorphisms between the spectra. $\newcommand{\bR}{\mathbb{R}}$
It may help to assume first that your topological space $(X,\tau)$ is compact, for then the space of   finite Borel measures can be identified with the space of   bounded continuous linear maps $C(X)\to\bR$. We denote $\newcommand{\eP}{\mathscr{P}}$ by $\eP(X)$ the space of  Borel probability measures on $X$. Then $\eP(X)$ is a closed convex subset of  the dual $C(X)^*$.   The map  
$$ X\ni x\mapsto \delta_x \in \eP(X) $$
maps $X$ bijectively  onto the set of extremal points of $\eP(X)$.  (Above $\delta_x$ denotes the Dirac $\delta$-measure supported at $x$.)  This map is continuous  (w.r.t. to the weak topology on $C(X)^*$) and thus establishes a homeomorphism between $X$ and the set of extremal points of $\eP(X)$. In this  sense $\eP(X)$ determines $X$ but this answer is  far less satisfactory   than Gelfand's theorem mentioned above.  
