  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 a
sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we
study the situation when the space $K$ admits a Josefson-Nissenzweig sequence
of measures such that its every element has finite support. We prove among the
others that $K$ admits such a Josefson-Nissenzweig sequence if and only if
$C(K)$ does not have the Grothendieck property restricted to functionals from
the space $\ell_1(K)$. We also investigate miscellaneous analytic and
topological properties of finitely supported Josefson-Nissenzweig sequences on
general Tychonoff spaces.
  We prove that various properties of compact spaces guarantee the existence of
finitely supported Josefson-Nissenzweig sequences. One such property is, e.g.,
that a compact space can be represented as the limit of an inverse system of
compact spaces based on simple extensions. An immediate consequence of this
result is that many classical consistent examples of Efimov spaces, i.e. spaces
being counterexamples to the famous Efimov problem, admit such sequences of
measures.
  Similarly, we show that if $K$ and $L$ are infinite compact spaces, then
their product $K\times L$ always admits a finitely supported
Josefson--Nissenzweig sequence. As a corollary we obtain a constructive proof
that 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 and
Freniche.
  Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for
the case of Banach spaces $C(K)$.
