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Sectionifying

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In [[mathematics]], the '''positive part''' of a [[real number|real]] or [[extended real number line|extended real]]-valued [[function (mathematics)|function]] is defined by the formula

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

Intuitively, the [[graph of a function|graph]] of <math>f^+</math> is obtained by taking the graph of <math>f</math>, chopping off the part under the ''x''-axis, and letting <math>f^+</math> take the value zero there.

Similarly, the '''negative part''' of ''f'' is defined as
:<math> f^-(x) = -\min(f(x),0) = \begin{cases} -f(x) & \mbox{ if } f(x) < 0 \\ 0 & \mbox{ otherwise.} \end{cases} </math>

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

:<math> f = f^+ - f^-. \, </math>

Also note that

:<math> |f| = f^+ + f^-\,</math>.

Using these two equations one may express the positive and negative parts as
:<math> f^+= \frac{|f| + f}{2}\,</math>
:<math> f^-= \frac{|f| - f}{2}.\,</math>

Another representation, using the [[Iverson bracket]] is

:<math> f^+= [f>0]f\,</math>
:<math> f^-= -[f<0]f.\,</math>

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

==Measure-theoretic properties==
Given a [[sigma-algebra|measurable space]] (''X'',Σ), an extended real-valued function ''f'' is [[measurable function|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
:<math>f=1_V-{1\over2},</math>
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==

*{{cite book
| last = Jones
| first = Frank
| title = Lebesgue integration on Euclidean space, Rev. ed
| publisher = Sudbury, Mass.: Jones and Bartlett
| date = 2001
| pages =
| isbn = 0-7637-1708-8
}}

*{{cite book
| last = Hunter
| first = John K
| coauthors = Nachtergaele, Bruno
| title = Applied analysis
| publisher = Singapore; River Edge, NJ: World Scientific
| date = 2001
| pages =
| isbn = 981-02-4191-7
}}

*{{cite book
| last = Rana
| first = Inder K
| title = An introduction to measure and integration, 2nd ed
| publisher = Providence, R.I.: American Mathematical Society
| date = 2002
| pages =
| isbn = 0-8218-2974-2
}}

==External links==

* [http://mathworld.wolfram.com/PositivePart.html Positive part] on [[MathWorld]]

[[Category:Elementary mathematics]]

Block-stacking problem - Revision history

en>Brirush: Sectionifying

en>David Eppstein: /* References */ authorlinks