Corollary 2

— Let $\theta$ be a complex measure on $X$; set $\theta_1 = (\Re \theta)^+$, $\theta_2 = (\Re \theta)^-$, $\theta_3 = (\Im \theta)^+$, $\theta_4 = (\Im \theta)^-$. In order that a mapping $f$ of $X$ into a topological space $G$ (resp. into a Banach space $F$) be measurable (resp. essentially integrable, integrable) for the measure $\theta$, it is necessary and sufficient that it be measurable (resp. essentially integrable, integrable) for each of the measures $\theta_i$ ($i = 1, 2, 3, 4$).

If $f$ is measurable (resp. essentially integrable, integrable) for $\theta$, then $f$ is by definition measurable (resp. essentially integrable, integrable) for the measure $|\theta|$, hence also for the measures $\theta_i$, which are $\leq |\theta|$. Conversely, if $f$ is measurable (resp. essentially integrable, integrable) for the measures $\theta_i$, then Prop. 2 (resp. Prop. 3, Cor. 1 of Prop. 3) implies that $f$ is measurable (resp. essentially integrable, integrable) for the measure $\theta_1 + \theta_2 + \theta_3 + \theta_4$, which is $\geq |\theta|$.