Definition 24.3.3 (Integral).label Let $(X, \cm, \mu)$ be a measure space and $f \in \mathcal{L}^{1}(X)$. If $f$ is $\real$-valued, then
\[\int f d\mu = \int f^{+} d\mu - \int f^{-} d\mu\]
is the integral of $f$. If $f$ is $\complex$-valued, then
\[\int f d\mu = \int \text{Re}(f)d\mu + i\int \text{Im}(f)d\mu\]
is the integral of $f$.
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