Positive and negative parts
In mathematics, the positive part of a real or extended real-valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of, 'chopping off' the part under the -axis, and letting take the value zero there.
Similarly, the negative part of is defined as
Note that both and are non-negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part.
The function can be expressed in terms of and as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
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, an extended real-valued function is measurable if and only if its positive and negative parts are. Therefore, if such a function is measurable, so is its absolute value, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking aswhere is a Vitali set, it is clear that 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.