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

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) ${\int }_{X}fd{\mu }_n\to 0$ for every continuous function $f\in C\left(X\right)$. 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) $\mathrm{supp}\left({\mu }_n\right)\cap \mathrm{supp}\left({\mu }_k\right)=\varnothing$ for every $n\ne k\in \omega$, and ii) the union ${\bigcup }_{n\in \omega }\mathrm{supp}\left({\mu }_n\right)$ 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 $\left|\mathrm{supp}\right({\mu }_n\left)\right|=2$ for every $n\in \omega$, or for every JN-sequence $〈{\mu }_n:n\in \omega 〉$ on X we have ${\lim }_{n\to \infty }\left|\mathrm{supp}\right({\mu }_n\left)\right|=\infty$.