Two countability properties of sets of measures (Q1074733)

For a Hausdorff space X, the spaces of the \(\sigma\)-additive, \(\tau\)- additive and tight Baire measures on X are denoted by \(M_{\sigma}(X)\), \(M_{\tau}(X)\) and \(M_ t(X)\). The positive cone of \(M_ s(X)\) is denoted by \(M^+_ s(X)\), for \(s=\sigma,\tau\) or t. The paper deals with the following countability properties: A subset M of \(M_{\sigma}(X)\) is called countably separated (c.s.) [resp. countably determined (c.d) in \(M_{\sigma}(X)]\) if there exists a sequence \(\{f_ n\}\) of bounded continuous real-valued functions on X such that for every \(\nu \in M\) the following holds: \(\mu \in M\) and \(\int f_ nd\mu =\int f_ nd\nu\) for all \(n\Rightarrow \mu =\nu [resp.\quad \mu \in M_{\sigma}(X)\quad and\quad \int f_ nd\mu =\int f_ nd\nu \quad for\quad all\quad n\Rightarrow \mu \in M].\) For a subset M of \(M^+_{\sigma}(X)\), the c.d. property of M in \(M^+_{\sigma}(X)\) is defined similarly. The main results are as follows. Let X be a compact space without isolated points and let \(M_ X\) denote the set of nonatomic measures in \(M^+_{\sigma}(X)\). Then the c.s. property of \(M_ X\) is equivalent to the weak separability and metrizability of \(M_ X\) and implies the c.d. property of \(M_ X\) in \(M^+_{\sigma}(X)\) and the separability of X. If, moreover, X is totally ordered all these conditions are equivalent but in general they are not. For arbitrary spaces X, several conditions equivalent to the c.s. property of \(M_ s(X)\), for \(s=\sigma,\tau\) or t, are given. It is also proved that if X is a metric space then \(M^+_{\tau}(X)\) is c.d. in \(M^+_{\sigma}(X)\) if and only if either \(card(X)\leq c\) or \(M^+_{\sigma}(X)= M^+_{\tau}(X).\)