Positive and negative parts

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

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 $$f^{+}(x)=\max (f(x),0)=\{\begin{array}{ll}f(x)& \text{ if }f(x)>0\\ 0& \text{ otherwise.}\end{array}$$ Intuitively, the graph of $f^{+}$ 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 $$f^{-}(x)=\max (-f(x),0)=-\min (f(x),0)=\{\begin{array}{ll}-f(x)& \text{ if }f(x)<0\\ 0& \text{ otherwise}\end{array}$$ 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 $$f=f^{+}-f^{-}.$$ Also note that $$|f|=f^{+}+f^{-}.$$ Using these two equations one may express the positive and negative parts as $$\begin{array}{rl}{f}^{+}& =\frac{|f|+f}{2}\\ f^{-}& =\frac{|f|-f}{2}.\end{array}$$ Another representation, using the Iverson bracket is $$\begin{array}{rl}{f}^{+}& =[f>0]f\\ f^{-}& =-[f<0]f.\end{array}$$ 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 $$f=1_V-\frac{1}{2},$$ 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

• Rectifier (neural networks)
• Even and odd functions
• Real and imaginary parts

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