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Given a Tychonoff space $X$, we call a sequence $\langle\mu_n\colonn\in\omega\rangle$ of signed Borel measures on $X$ a finitely supportedJosefson--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 measuresand $\|\mu_n\|=1$, and 2) $\int_Xf\,\mathrm{d}\mu_n\to0$ for every continuousfunction $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$admits a JN-sequence, then there exists a JN-sequence $\langle\mu_n\colonn\in\omega\rangle$ such that: i)$\mbox{supp}(\mu_n)\cap\mbox{supp}(\mu_k)=\emptyset$ for every $n\neqk\in\omega$, and ii) the union $\bigcup_{n\in\omega}\mbox{supp}(\mu_n)$ is adiscrete subset of $X$. We also prove that if a Tychonoff space $X$ carries aJN-sequence, then either there is a JN-sequence $\langle\mu_n\colonn\in\omega\rangle$ on $X$ such that $|\mbox{supp}(\mu_n)|=2$ for every$n\in\omega$, or for every JN-sequence $\langle\mu_n\colon n\in\omega\rangle$on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(\mu_n)|=\infty$.

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