Editing Lebesgue measure (section)

== Construction of the Lebesgue measure ==
The modern construction of the Lebesgue measure is an application of [[Carathéodory's extension theorem]]. It proceeds as follows.

Fix <math>n \in \mathbb N</math>. A '''box''' in <math>\mathbb{R}^n</math> is a set of the form<math display="block">B=\prod_{i=1}^n [a_i,b_i] \, ,</math>where <math>b_i \geq a_i</math>, and the product symbol here represents a Cartesian product. The volume of this box is defined to be<math display="block">\operatorname{vol}(B)=\prod_{i=1}^n (b_i-a_i) \, .</math>For ''any'' subset ''<math>A</math>'' of <math>\mathbb{R}^n</math>, we can define its [[outer measure]] <math>\lambda^{\!*\!}(A)</math> by:<math display="block">\lambda^*(A) = \inf \left\{\sum_{B\in \mathcal{C}}\operatorname{vol}(B) : \mathcal{C}\text{ is a countable collection of boxes whose union covers }A\right\} .</math>We then define the set ''<math>A</math>'' to be Lebesgue-measurable if for every subset ''<math>S</math>'' of <math>\mathbb{R}^n</math>,<math display="block">\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S \setminus A) \, .</math>These Lebesgue-measurable sets form a [[σ-algebra|''σ''-algebra]], and the Lebesgue measure is defined by <math>\lambda(A) = \lambda^{\!*\!}(A)</math> for any Lebesgue-measurable set ''<math>A</math>''.

The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical [[axiom of choice]], which is independent from many of the conventional systems of axioms for [[set theory]]. The [[Vitali set|Vitali theorem]], which follows from the axiom, states that there exist subsets of '''<math>\mathbb{R}</math>''' that are not Lebesgue-measurable. Assuming the axiom of choice, [[non-measurable set]]s with many surprising properties have been demonstrated, such as those of the [[Banach–Tarski paradox]].

In 1970, [[Robert M. Solovay]] showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework of [[Zermelo–Fraenkel set theory]] in the absence of the axiom of choice (see [[Solovay's model]]).<ref>{{Cite journal |last=Solovay |first=Robert M. |title=A model of set-theory in which every set of reals is Lebesgue-measurable |journal=[[Annals of Mathematics]] |jstor=1970696 |series=Second Series |volume=92 |year=1970 |issue=1 |pages=1–56 |doi=10.2307/1970696 }}</ref>