Positive and negative parts

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

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

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^{-}(x)=\max(-f(x),0)=-\min(f(x),0)={\begin{cases}-f(x)&{\text{ if }}f(x)<0\\0&{\text{ 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

f=f^{+}-f^{-}.

Also note that

|f|=f^{+}+f^{-}.

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

{\begin{aligned}f^{+}&={\frac {|f|+f}{2}}\\f^{-}&={\frac {|f|-f}{2}}.\end{aligned}}

Another representation, using the Iverson bracket is

{\begin{aligned}f^{+}&=[f>0]f\\f^{-}&=-[f<0]f.\end{aligned}}

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

The unit ramp function is the positive part of the identity function.

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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}-{\frac {1}{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.

References

Jones, Frank (2001). Lebesgue integration on Euclidean space (Rev. ed.). Sudbury, MA: Jones and Bartlett. ISBN 0-7637-1708-8.
Hunter, John K; Nachtergaele, Bruno (2001). Applied analysis. Singapore; River Edge, NJ: World Scientific. ISBN 981-02-4191-7.
Rana, Inder K (2002). An introduction to measure and integration (2nd ed.). Providence, R.I.: American Mathematical Society. ISBN 0-8218-2974-2.

External links

Positive part on MathWorld

This page was last edited on 17 February 2024, at 23:53

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Measure-theoretic properties

See also

References

External links