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In mathematics, the positive part of a real or extended real-valued function is defined by the formula## 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
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- $f^+(x)\; =\; max(f(x),0)\; =\; begin\{cases\}\; f(x)\; \&\; mbox\{\; if\; \}\; f(x)\; >\; 0\; 0\; \&\; mbox\{\; otherwise.\}\; end\{cases\}$

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

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

where the vertical bars denote the absolute value.

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.

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

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.

- 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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Last updated on Sunday April 29, 2007 at 07:28:21 PDT (GMT -0700)

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