Integrable Function and the Lebesgue Integral

Let $(X,\mathcal{F},\mu)$ be a measure space and let $f:X\to\mathbb{R}$ be measurable. Define the positive part $f^{+}$ and negative part $f^{-}$ by

so that $f=f^{+}-f^{-}$ and $|f|=f^{+}+f^{-}$. Both $f^{+}$ and $f^{-}$ are measurable: for $a\ge 0$, $\{f^{+}>a\}=\{f>a\}$ and $\{f^{-}>a\}=\{f<-a\}$, while for $a<0$ both sets equal $X$; the criterion of Measurable Function and Real-Valued Measurable Function applies.

The function $f$ is integrable (with respect to $\mu$) if both $\int_X f^{+}\,d\mu$ and $\int_X f^{-}\,d\mu$ are finite, the integrals being those of Lebesgue Integral of a Nonnegative Measurable Function; in that case the Lebesgue integral of $f$ is

By the additivity of the nonnegative integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), $f$ is integrable if and only if $\int_X|f|\,d\mu<\infty$.