Positive part

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. {\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\text{ if }}f(x)>0\\0&{\text{ otherwise.}}\end{cases}}}

Intuitively, the graph of f + {\displaystyle f^{+}} is obtained by taking the graph of f {\displaystyle f} , 'chopping off' the part under the x-axis, and letting f + {\displaystyle f^{+}} take the value zero there.

Similarly, the negative part of f is defined as f − ( x ) = max ( − f ( x ) , 0 ) = − min ( f ( x ) , 0 ) = { − f ( x )  if  f ( x ) < 0 0  otherwise {\displaystyle f^{-}(x)=\max(-f(x),0)=-\min(f(x),0)={\begin{cases}-f(x)&{\text{ if }}f(x)<0\\0&{\text{ otherwise}}\end{cases}}}

Note that both f⁺ and f⁻ are non-negative functions. A peculiarity of terminology is that the 'negative part' is not negative (like the imaginary part of a complex number is not imaginary).

The function f can be expressed in terms of f⁺ and f⁻ as f = f + − f − . {\displaystyle f=f^{+}-f^{-}.}

Also note that | f | = f + + f − . {\displaystyle |f|=f^{+}+f^{-}.}

Using these two equations one may express the positive and negative parts as f + = | f | + f 2 f − = | f | − f 2 . {\displaystyle {\begin{aligned}f^{+}&={\frac {|f|+f}{2}}\\f^{-}&={\frac {|f|-f}{2}}.\end{aligned}}}

Another representation, using the Iverson bracket is f + = [ f > 0 ] f f − = − [ f < 0 ] f . {\displaystyle {\begin{aligned}f^{+}&=[f>0]f\\f^{-}&=-[f<0]f.\end{aligned}}}

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 |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 , {\displaystyle f=1_{V}-{\frac {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.

See also

• Rectifier (neural networks)
• Even and odd functions
• Real and imaginary parts

References

External links

• Positive part on MathWorld