 Positive part of a real-valued function

The positive part of a real-valued function $f$ is the nonnegative function $f_{+}(x)=\max \{0,f(x)\}$. It retains positive values and replaces negative values by zero: if $f(x)=-3$, then $f_{+}(x)=0$, while $f(x)=2$ gives $f_{+}(x)=2$.

Together with the negative part $f_{-}(x)=\max \{0,-f(x)\}$, it satisfies $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$. These identities reduce Lebesgue integration of a signed function to integration of nonnegative functions. The resulting difference of integrals is defined only when it does not subtract two infinite quantities; integrability requires both integrals to be finite.