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The celebrated Josefson-Nissenzweig theorem implies that for a Banach space$C(K)$ of continuous real-valued functions on an infinite compact space $K$there exists a sequence of Radon measures $\langle\mu_n\colon\n\in\omega\rangle$ on $K$ which is weakly* convergent to the zero measure on$K$ and such that $\big\|\mu_n\big\|=1$ for every $n\in\omega$. We call such asequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper westudy the situation when the space $K$ admits a Josefson-Nissenzweig sequenceof measures such that its every element has finite support. We prove among theothers that $K$ admits such a Josefson-Nissenzweig sequence if and only if$C(K)$ does not have the Grothendieck property restricted to functionals fromthe space $\ell_1(K)$. We also investigate miscellaneous analytic andtopological properties of finitely supported Josefson-Nissenzweig sequences ongeneral Tychonoff spaces.  We prove that various properties of compact spaces guarantee the existence offinitely supported Josefson-Nissenzweig sequences. One such property is, e.g.,that a compact space can be represented as the limit of an inverse system ofcompact spaces based on simple extensions. An immediate consequence of thisresult is that many classical consistent examples of Efimov spaces, i.e. spacesbeing counterexamples to the famous Efimov problem, admit such sequences ofmeasures.  Similarly, we show that if $K$ and $L$ are infinite compact spaces, thentheir product $K\times L$ always admits a finitely supportedJosefson--Nissenzweig sequence. As a corollary we obtain a constructive proofthat the space $C_p(K\times L)$ contains a complemented copy of the space $c_0$endowed with the pointwise topology--this generalizes results of Cembranos andFreniche.  Finally, we provide a direct proof of the Josefson-Nissenzweig theorem forthe case of Banach spaces $C(K)$.

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces