mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the positive part of a real or extended real-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) & \text f(x) > 0 \\
0 & \text
\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 discret ...

f^+

is obtained by taking the graph of

f

, 'chopping off' the part under the -axis, and letting

f^+

take the value zero there. Similarly, the negative part of is defined as

f^-(x) = \max(-f(x),0) = -\min(f(x),0) = \begin
-f(x) & \text f(x) < 0 \\
0 & \text
\end

Note that both and 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 can be expressed in terms of and as

f = f^+ - f^-.

Also note that

, f,  = f^+ + f^-.

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

\begin
f^+ &= \frac \\
f^- &= \frac.
\end

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. ...

. \end

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 (mathematics), group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G ...

. 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 ...

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. It captures and generalises intuitive notions such as length, area, an ...

, an extended real-valued function 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 magnitude, mass, and probability of events. These seemingly distinct concepts hav ...

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

its positive and negative parts are. Therefore, if such a function is measurable, so is its absolute value , being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking as

f = 1_V - \frac,

where is a

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

, it is clear that 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 (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

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

signed measure In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign. Definition There are two slightly different concepts of a signed measure, de ...

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 -mea ...

References

* * * {{refend

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