, the positive part
of a real
or extended real
is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x-axis, and letting take the value zero there.
Similarly, the negative part of f is defined as
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
Also note that
where the vertical bars denote the absolute value.
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.
Given a measurable space
,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
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.
- 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.