Positive and negative parts - Alchetron, the free social encyclopedia

In mathematics, the positive part of a real or extended real-valued function is defined by the formula

f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise.

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 ) = − min ( f ( x ) , 0 ) = { − f ( x ) if f ( x ) < 0 0 otherwise.

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

f = f + − f − .

Also note that

| f | = f + + f − .

Using these two equations one may express the positive and negative parts as

f + = | f | + f 2 f − = | f | − f 2 .

Another representation, using the Iverson bracket is

f + = [ f > 0 ] f f − = − [ f < 0 ] f .

One may define the positive and negative part of any function with values in a linearly ordered group.

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 |f|, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking f as

f = 1 V − 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.

References

Positive and negative parts Wikipedia

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