On extension of $f$-divergence

For a lower semicontinuous convex function $f:\mathbf{R}\to\mathbf{R}\cup\{+\infty\}$, $\mathrm{dom}\,f\subseteq\mathbf{R}_+$, we give a definition and study properties of the $f$-divergence of finitely additive set functions $\mu$ and $\nu$ given on a measurable space $(\Omega,\mathscr{F})$. If $f$ is finite on $(0,+\infty)$ and $\mu$ and $\nu$ are probability measures, our definition is equivalent to the classical definition of the $f$-divergence introduced by Csiszár. As an application, we obtain a result on attaining the minimum by the $f$-divergence over a set $\mathscr{Z}$ of pairs of probability measures.