Let $(E,\mathcal E)$ be a measurable space, let $\mu:\mathcal E\to[0,\infty]$ be a positive measure, and let $\nu:\mathcal E\to\mathbb C$ be a finite complex measure. Denote by $|\nu|:\mathcal E\to[0,\infty)$ the total variation measure of $\nu$. Then $\nu\ll\mu$ if and only if for every $\varepsilon>0$ there exists $\delta>0$ such that, for every $A\in\mathcal E$, $\mu(A)<\delta$ implies $|\nu|(A)<\varepsilon$.