Block-stacking problem

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

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

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

$f^{-}(x)=-\min (f(x),0)=\{\begin{array}{ll}-f(x)& \text{ if }f(x)<0\\ 0& \text{ otherwise.}\end{array}$

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

$\left|f\right|=f^{+}+f^{-}$.

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

$f^{+}=\frac{\left|f\right|+f}{2}$

$f^{-}=\frac{\left|f\right|-f}{2}.$

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

$f=1_V-\frac{1}{2},$

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.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Positive part on MathWorld