 Negative part of a real-valued function

The negative part of a real-valued function $f$ is the nonnegative function $f_{-}(x)=\max (0,-f(x))$. It records the magnitude of the negative values, rather than retaining their sign: if $f(x)=-3$, then $f_{-}(x)=3$. Together with the positive part, it gives $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$. This decomposition defines the Lebesgue integral of a signed function and makes the integrability condition explicit.