On points avoiding measures

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 $\omega$

Sequential closure in the space of measures

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.Comment: (18 pages, a gap in an argument from the previous version fixed

On the spaces dual to combinatorial Banach spaces

We present quasi-Banach spaces which are isomorphic to the duals of combinatorial Banach spaces and which are equipped with quasi-norms that seem to be much easier to handle than the original dual norms. In particular we show nice quasi-renormings of the dual Schreier spaces. We use it to show that the duals to the combinatorial Banach spaces induced by large families (in the sense of Lopez-Abad and Todorcevic) are $\ell_1$-saturated and do not have the Schur property.Comment: 19 page