# Positive and negative parts

In mathematics, the positive part of a real or extended real-valued function is defined by the formula

${\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\mbox{ if }}f(x)>0\\0&{\mbox{ otherwise.}}\end{cases}}}$

Intuitively, the graph of ${\displaystyle f^{+}}$ is obtained by taking the graph of ${\displaystyle f}$, chopping off the part under the x-axis, and letting ${\displaystyle f^{+}}$ take the value zero there.

Similarly, the negative part of f is defined as

${\displaystyle f^{-}(x)=\max(-f(x),0)=-\min(f(x),0)={\begin{cases}-f(x)&{\mbox{ if }}f(x)<0\\0&{\mbox{ otherwise.}}\end{cases}}}$

Note that both f+ and f are non-negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part).

The function f can be expressed in terms of f+ and f as

${\displaystyle f=f^{+}-f^{-}.}$

Also note that

${\displaystyle |f|=f^{+}+f^{-}}$.

Using these two equations one may express the positive and negative parts as

${\displaystyle f^{+}={\frac {|f|+f}{2}}}$
${\displaystyle f^{-}={\frac {|f|-f}{2}}.}$

Another representation, using the Iverson bracket is

${\displaystyle f^{+}=[f>0]f}$
${\displaystyle f^{-}=-[f<0]f.}$

One may define the positive and negative part of any function with values in a linearly ordered group.

## Measure-theoretic properties

Given a measurable space (X,Σ), an extended real-valued function f is measurable if and only if its positive and negative parts are. Therefore, if such a function f is measurable, so is its absolute value |f|, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking f as

${\displaystyle f=1_{V}-{1 \over 2},}$

where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function.

The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.