On sequences of finitely supported measures related to the Josefson--Nissenzweig theorem

Abstract

Given a Tychonoff space $X$, we call a sequence $⟨{\mu }_n:n\in \omega ⟩$ of signed Borel measures on $X$ a finitely supported Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every $n\in \omega$ the measure ${\mu }_n$ is a finite combination of one-point measures and $\Vert {\mu }_n\Vert =1$, and 2) ${\int }_{X}f\mathrm{d}{\mu }_n\to 0$ for every continuous function $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$ admits a JN-sequence, then there exists a JN-sequence $⟨{\mu }_n:n\in \omega ⟩$ such that: i) $\mbox{supp}(μ_n)\cap\mbox{supp}(μ_k)=\emptyset$ for every $n\ne k\in \omega$, and ii) the union $\bigcup_{n\inω}\mbox{supp}(μ_n)$ is a discrete subset of $X$. We also prove that if a Tychonoff space $X$ carries a JN-sequence, then either there is a JN-sequence $⟨{\mu }_n:n\in \omega ⟩$ on $X$ such that $|\mbox{supp}(μ_n)|=2$ for every $n\in \omega$, or for every JN-sequence $⟨{\mu }_n:n\in \omega ⟩$ on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(μ_n)|=\infty$.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2009.07552