Definitions

# Positive and negative parts

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

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

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

Similarly, the negative part of f is defined as

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

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

$f = f^+ - f^-. ,$

Also note that

$|f| = f^+ + f^-,$

where the vertical bars denote the absolute value.

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

$f^+= frac\left\{|f| + f\right\}\left\{2\right\},$
$f^-= frac\left\{|f| - f\right\}\left\{2\right\}.,$

Another representation, using the Iverson bracket is

$f^+= \left[f>0\right]f,$
$f^-= -\left[f<0\right]f.,$

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
$f=1_V-\left\{1over2\right\},$
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.

## References

• Jones, Frank Lebesgue integration on Euclidean space, Rev. ed. Sudbury, Mass.: Jones and Bartlett.
• Hunter, John K; Nachtergaele, Bruno Applied analysis. Singapore; River Edge, NJ: World Scientific.
• Rana, Inder K An introduction to measure and integration, 2nd ed. Providence, R.I.: American Mathematical Society.

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