Positive and negative parts

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

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

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

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

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

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

Also note that

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

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

$f^+= \frac{|f| + f}{2}\,$

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

Another representation, using the Iverson bracket is

$f^+= [f>0]f\,$

$f^-= -[f<0]f.\,$

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

$f=1_V-{1\over2},$

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, Mass.: Jones and Bartlett. ISBN 0763717088. 

• Hunter, John K; Nachtergaele, Bruno (2001). Applied analysis. Singapore; River Edge, NJ: World Scientific. ISBN 9810241917. 

• Rana, Inder K (2002). An introduction to measure and integration, 2nd ed. Providence, R.I.: American Mathematical Society. ISBN 0821829742. 

External links

• Positive part on MathWorld