Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$f : X \to \mathbb{C}$ is a D5617: Complex Borel function on $M$
(ii)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(iii)	\begin{equation} \mu(E) = 0 \end{equation}

Since we define \begin{equation} \int_E f \, d \mu = \int_E \Re f \, d \mu + i \int_E \Im f \, d \mu \end{equation} then this result follows as a consequence of the result R4562: Basic integral over a set of measure zero is zero. $\square$