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Positive and negative parts

Decomposition of real-valued functions

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated August 10, 2026
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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)&{\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 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{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 not negative (like the imaginary part of a complex number is not imaginary).

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

Another representation, using the Iverson bracket is {\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.

01Measure-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.

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This article is adapted from the Wikipedia article “Positive and negative parts”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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