In mathematics, the positive part of a real or extended real-valued function is defined by the formula
Intuitively, the graph of
f+
f
f+
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 (like the imaginary part of a complex number is neither imaginary 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
| + f |
| - f |
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.
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.