On points avoiding measures

Abstract

We say that an element x of a topological space X avoids measures if for every Borel measure μ on X if $\mu \left(\right\{x\left\}\right)=0$, then there is an open $U\ni x$ such that $\mu \left(U\right)=0$. The negation of this property can viewed as a local version of the property of supporting a strictly positive measure. We study points avoiding measures in the general setting as well as in the context of ${\omega }^{⁎}$, the remainder of Stone-Čech compactification of ω.