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Definition D1749
Complex integral

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 $$\int_X f \, d \mu : = \int_X \Re (f) \, d \mu + i \int_X \Im (f) \, d \mu$$

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 $$\int_X f \, d \mu : = \bigg( \int_X \Re f \, d \mu, \int_X \Im f \, d \mu \bigg)$$

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 $$\mu(f) : = \mu(\Re f) + i \mu(\Im f)$$
Results
 ▶ R4573: Complex expectation over a null event is zero ▶ R4563: Complex integral over a set of measure zero is zero