= Positive part of a real-valued function
{title2=$f_+$}
= Positive part
{synonym}
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.
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