In mathematics, the positive part of a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...

extended real In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...

-valued

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

is defined by the formula :

f^+(x) = \max(f(x),0) = \begin f(x) & \mbox f(x) > 0 \\ 0 & \mbox \end

Intuitively, the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

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 -f(x) & \mbox f(x) < 0 \\ 0 & \mbox \end

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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

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^-= \frac.

Another representation, using the

Iverson bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement ...

is :

f^+= >0

f^-= - <0 .

One may define the positive and negative part of any function with values in a

linearly ordered group In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a: * l ...

. The unit

ramp function The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for ot ...

is the positive part of the

identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

Measure-theoretic properties

Given a

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then ...

(''X'',Σ), an extended real-valued function ''f'' is

measurable In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

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-,

where ''V'' is a

Vitali set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vi ...

, 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 In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...

for a real-valued function. Analogously to this decomposition of a function, one may decompose a

signed measure In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not ...

into positive and negative parts — see the

Hahn decomposition theorem In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space (X,\Sigma) and any signed measure \mu defined on the \sigma -algebra \Sigma , there exist two \Sigma -m ...

References

* * *{{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

Positive part
on

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

Elementary mathematics

Measure-theoretic properties

See also

References

External links