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positive and negative parts

{{Short description|Decomposition of real-valued functions}}

File:Positive and Negative Parts of f(x) = x^2 - 4.png

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) & \text{ if } f(x) > 0 \\

0 & \text{ otherwise.}

\end{cases}

Intuitively, the graph of f^+ is obtained by taking the graph of f, 'chopping off' the part under the {{math|x}}-axis, and letting f^+ take the value zero there.

Similarly, the negative part of {{math|f}} is defined as

f^-(x) = \max(-f(x),0) = -\min(f(x),0) = \begin{cases}

-f(x) & \text{ if } f(x) < 0 \\

0 & \text{ otherwise}

\end{cases}

Note that both {{math|f+}} and {{math|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 {{math|f}} can be expressed in terms of {{math|f+}} and {{math|f−}} as

f = f^+ - f^-.

Also note that

|f| = f^+ + f^-.

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

\begin{align}

f^+ &= \frac{|f| + f}{2} \\

f^- &= \frac{|f| - f}{2}.

\end{align}

Another representation, using the Iverson bracket is

\begin{align}

f^+ &= [f>0]f \\

f^- &= -[f<0]f.

\end{align}

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

The unit ramp function is the positive part of the identity function.

Measure-theoretic properties

Given a measurable space {{math|(X, Σ)}}, an extended real-valued function {{math|f}} is measurable if and only if its positive and negative parts are. Therefore, if such a function {{math|f}} is measurable, so is its absolute value {{math|{{abs|f}}}}, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking {{math|f}} as

f = 1_V - \frac{1}{2},

where {{math|V}} is a Vitali set, it is clear that {{math|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.

See also

References

{{refbegin}}

  • {{cite book

| last = Jones

| first = Frank

| title = Lebesgue integration on Euclidean space

| edition = Rev.

| location = Sudbury, MA

| publisher = Jones and Bartlett

| date = 2001

| pages =

| isbn = 0-7637-1708-8

}}

  • {{cite book

| last1 = Hunter

| first1 = John K

| last2 = Nachtergaele

| first2 = 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

| edition = 2nd

| location = Providence, R.I.

| publisher = American Mathematical Society

| date = 2002

| pages =

| isbn = 0-8218-2974-2

}}

{{refend}}

External links

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

Category:Elementary mathematics