Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $f : X \to \mathbb{C}$ be an D1921: Absolutely integrable function on $M$.
The integral of $f$ with respect to $M$ is the D1207: Complex number \begin{equation} \int_X f \, d \mu : = \int_X \Re (f) \, d \mu + i \int_X \Im (f) \, d \mu \end{equation}

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $f : X \to \mathbb{C}$ be an D1921: Absolutely integrable function on $M$.
The integral of $f$ with respect to $M$ is the D1207: Complex number \begin{equation} \int_X f \, d \mu : = \bigg( \int_X \Re f \, d \mu, \int_X \Im f \, d \mu \bigg) \end{equation}

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $f : X \to \mathbb{C}$ be an D1921: Absolutely integrable function on $M$.
The integral of $f$ with respect to $M$ is the D1207: Complex number \begin{equation} \mu(f) : = \mu(\Re f) + i \mu(\Im f) \end{equation}