Positive and negative parts

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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) & \mbox{ if } f(x) > 0 \\ 0 & \mbox{ otherwise.} \end{cases}

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) = -\min(f(x),0) = \begin{cases} -f(x) & \mbox{ if } f(x) < 0 \\ 0 & \mbox{ 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^-. \,

Also note that

 |f| = f^+ + f^-\,.

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

 f^+= \frac{|f| + f}{2}\,
 f^-= \frac{|f| - f}{2}.\,

Another representation, using the Iverson bracket is

 f^+= [f>0]f\,
 f^-= -[f<0]f.\,

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

Measure-theoretic properties[edit]

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\over2},

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.

References[edit]

  • Jones, Frank (2001). Lebesgue integration on Euclidean space, Rev. ed. Sudbury, Mass.: 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. 

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