Baire category in spaces of measures (Q1354653)

Given a topological space \(X\), let \(M(X)\) denote the space of non-negative finite Borel measures on X. The weak topology on \(M(X)\) is the smallest topology such that the function \(M(X)\ni \mu\to\mu(G)\) is lower semicontinuous for every open subset \(G\) of \(X\) and the function \(M(X)\ni \mu\to\mu(X)\) is continuous. It is known that if X is completely regular then the relative weak topology on the space \(M_t(X)\) of tight (or Radon) measures on \(X\) coincides with the weak topology induced by the space of bounded real-valued continuous functions on \(X\). As usual, \((\mu\times\mu)^*\) denotes the outer measure induced by \(\mu\times \mu\) and defined on all subsets of \(X\times X\). The main result of the paper under review is the following theorem: Let \(X\) be a Hausdorff space and \(R\) be a subset of \(X\times X\) of the first category. Then \((\mu\times \mu)^*(R)=0\) for all \(\mu\in M(X)\) except for a set of measures of the first category in \(M(X)\). Several interesting consequences of the above result to e.g., invariant sets of measures or on category zero-one laws are given.