Positive and negative parts

Short description: Decomposition of real-valued functions

Positive and Negative Parts of f(x) = x2 − 4

In mathematics, the positive part of a real or extended real-valued function is defined by the formula f+(x)=max⁡(f(x),0)={f(x) if f(x)>00 otherwise.

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)={−f(x) if f(x)<00 otherwise

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 f+=|f|+f2f−=|f|−f2.

Another representation, using the Iverson bracket is f+=[f>0]ff−=−[f<0]f.

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.

## Contents

* 1 Measure-theoretic properties
* 2 See also
* 3 References
* 4 External links

## 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=1V−12, 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.

## See also

* Rectifier (neural networks)
* Even and odd functions
* Real and imaginary parts

## 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

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