Positive part

Positive and negative parts

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^-,

where the vertical bars denote the absolute value.

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

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-{1over2},
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

  • Jones, Frank Lebesgue integration on Euclidean space, Rev. ed. Sudbury, Mass.: Jones and Bartlett.
  • Hunter, John K; Nachtergaele, Bruno Applied analysis. Singapore; River Edge, NJ: World Scientific.
  • Rana, Inder K An introduction to measure and integration, 2nd ed. Providence, R.I.: American Mathematical Society.

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