Decomposition of real-valued functions
In mathematics, the positive part of a real or extended real-valued function is defined by the formula
formula_1
Intuitively, the graph of formula_2 is obtained by taking the graph of formula_3, chopping off the part under the "x"-axis, and letting formula_2 take the value zero there.
Similarly, the negative part of "f" is defined as 
formula_5
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
formula_6
Also note that
formula_7
Using these two equations one may express the positive and negative parts as
formula_8
Another representation, using the Iverson bracket is
formula_9
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 ("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 , being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking "f" as
formula_10
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.
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