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

Given a Tychonoff space $X$, we call a sequence $⟨{\mu }_n\text{\hspace{0.17em}⁣}:n\in \omega ⟩$ 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 $\mathrm{\parallel }{\mu }_n\mathrm{\parallel }=1$, 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 $⟨{\mu }_n\text{\hspace{0.17em}⁣}:n\in \omega ⟩$ such that: i) $supp({\mu }_n)\text{capsupp}({\mu }_k)=\mathrm{\varnothing }$ for every $n\mathrm{\ne }k\in \omega$, and ii) the union ${\bigcup }_{n\in \omega }supp({\mu }_n)$ 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 $⟨{\mu }_n\text{\hspace{0.17em}⁣}:n\in \omega ⟩$ on $X$ such that $\mathrm{\mid }supp({\mu }_n)\mathrm{\mid }=2$ for every $n\in \omega$, or for every JN-sequence $⟨{\mu }_n\text{\hspace{0.17em}⁣}:n\in \omega ⟩$ on $X$ we have ${\lim }_{n\to \mathrm{\infty }}\mathrm{\mid }supp({\mu }_n)\mathrm{\mid }=\mathrm{\infty }$.