  We say that an element $x$ of a topological space $X$ avoids measures if for
every Borel measure $\mu$ on $X$ if $\mu(\{x\})=0$, then there is an open $U\ni
x$ such that $\mu(U)=0$. 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
$\omega^\ast$, the remainder of Stone-\v{C}ech compactification of $\omega$.
