Positive and negative parts

Decomposition of real-valued functions
Positive and Negative Parts of f(x) = x2 − 4

In mathematics, the positive part of a real or extended real-valued function is defined by the formula

f + ( x ) = max ( f ( x ) , 0 ) = { f ( x )  if  f ( x ) > 0 0  otherwise. {\displaystyle 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 + {\displaystyle f^{+}} is obtained by taking the graph of f {\displaystyle f} , chopping off the part under the x-axis, and letting f + {\displaystyle 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 ) = { − f ( x )  if  f ( x ) < 0 0  otherwise. {\displaystyle 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 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 − . {\displaystyle f=f^{+}-f^{-}.}

Also note that

| f | = f + + f − . {\displaystyle |f|=f^{+}+f^{-}.}

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

f + = | f | + f 2 f − = | f | − f 2 . {\displaystyle {\begin{aligned}f^{+}&={\frac {|f|+f}{2}}\\f^{-}&={\frac {|f|-f}{2}}.\end{aligned}}}

Another representation, using the Iverson bracket is

f + = [ f > 0 ] f f − = − [ f < 0 ] f . {\displaystyle {\begin{aligned}f^{+}&=[f>0]f\\f^{-}&=-[f<0]f.\end{aligned}}}

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 (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

f = 1 V − 1 2 , {\displaystyle f=1_{V}-{\frac {1}{2}},}
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.

See also

References

  • Jones, Frank (2001). Lebesgue integration on Euclidean space (Rev. ed.). Sudbury, MA: Jones and Bartlett. ISBN 0-7637-1708-8.
  • Hunter, John K; Nachtergaele, Bruno (2001). Applied analysis. Singapore; River Edge, NJ: World Scientific. ISBN 981-02-4191-7.
  • Rana, Inder K (2002). An introduction to measure and integration (2nd ed.). Providence, R.I.: American Mathematical Society. ISBN 0-8218-2974-2.

External links

  • Positive part on MathWorld