Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space} and let $f:X\to\mathbb{R}$ be \reftext{def:measurable-function-2026a}{measurable}. Define the \textbf{positive part} $f^{+}$ and \textbf{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 \ref{def:measurable-function-2026a} applies.

The function $f$ is \textbf{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 \ref{def:lebesgue-integral-nonnegative-2026a}; in that case the \textbf{Lebesgue integral} of $f$ is

By the additivity of the nonnegative integral (claim 1 of \ref{thm:linearity-monotonicity-integral-2026a}), $f$ is integrable if and only if $\int_X|f|\,d\mu<\infty$.