Dense sets of measures (Q1059727)

Let X be a topological space, \(M_{[+]}(X)\) the space of complete positive Baire measures on X with the weak topology \(\sigma (M(X),C_ b(X)),\) where \(C_ b(X)\) is the space of all bounded real-valued continuous functions on X. A subset \(B\subset M(X)\) is called dense if it is bounded and the corresponding majoration is uniform with respect to \(\mu\in B\). In some sense the following result is the best possible: If X is a metric space then each compact denumerable subset of \(M_+(X)\) is dense (Theorem 1). The second main result of this paper (Theorem 5) expresses the fact that on a good space there are many dense sets of measures. If X is functionally separable and each Baire measure is complete, then for every cylindrical measure \(\lambda\) on M(X) and \(\epsilon >0\), there exists a dense \(A\subset M(X)\) such that \(| \lambda |_*(M(X)\setminus A)<\epsilon.\)