Editing Lebesgue measure (section)

==Definition==

For any [[Interval (mathematics)|interval]] <math>I = [a,b]</math>, or <math>I = (a, b)</math>, in the set <math>\mathbb{R}</math> of real numbers, let <math>\ell(I)= b - a</math> denote its length. For any subset <math>E\subseteq\mathbb{R}</math>, the Lebesgue [[outer measure]]<ref>{{cite book |title=Real Analysis |last1=Royden |first1=H. L. |author-link=Halsey Royden |date=1988 |publisher=Macmillan |isbn=0-02-404151-3 |edition=3rd |location=New York |page=56 }}</ref> <math>\lambda^{\!*\!}(E)</math> is defined as an [[infimum]]

<math display="block">\lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \ell(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of open intervals with } E\subset \bigcup_{k=1}^\infty I_k\right\}.</math>

The above definition can be generalised to higher dimensions as follows.<ref>{{Cite web|url=https://de.wikipedia.org/w/index.php?title=Lebesgue-Ma%C3%9F&oldid=225731376|title=Lebesgue-Maß|date=29 August 2022|accessdate=9 March 2023|via=Wikipedia}}</ref>
For any [[rectangular cuboid]] <math>C</math> which is a [[Cartesian product]] <math>C=I_1\times\cdots\times I_n</math> of open intervals, let <math>\operatorname{vol}(C)=\ell(I_1)\times\cdots\times \ell(I_n)</math> (a real number product) denote its volume.
For any subset <math>E\subseteq\mathbb{R^n}</math>,

<math display="block">\lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \operatorname{vol}(C_k) : {(C_k)_{k \in \mathbb N}} \text{ is a sequence of products of open intervals with } E\subset \bigcup_{k=1}^\infty C_k\right\}.</math>

Some sets <math>E</math> satisfy the [[Carathéodory's criterion|Carathéodory criterion]], which requires that for every <math> A\subseteq \mathbb{R^n}</math>,

<math display="block">\lambda^{\!*\!}(A) = \lambda^{\!*\!}(A \cap E) + \lambda^{\!*\!}(A \cap E^c).</math>

Here <math>E^c</math> denotes the complement set. The sets <math>E</math> that satisfy the Carathéodory criterion are said to be Lebesgue-measurable, with its Lebesgue measure being defined as its Lebesgue outer measure: <math>\lambda(E) = \lambda^{\!*\!}(E)</math>. The set of all such <math>E</math> forms a [[Sigma-algebra|''σ''-algebra]]. 

A set <math>E</math> that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. [[ZFC]] proves that [[non-measurable set]]s do exist; examples are the [[Vitali set]]s.

=== Intuition ===

The first part of the definition states that the subset <math>E</math> of the real numbers is reduced to its outer measure by coverage by sets of open intervals.  Each of these sets of intervals <math>I</math> covers <math>E</math> in a sense, since the union of these intervals contains <math>E</math>. The total length of any covering interval set may overestimate the measure of <math>E,</math> because <math>E</math> is a subset of the union of the intervals, and so the intervals may include points which are not in <math>E</math>. The Lebesgue outer measure emerges as the [[Infimum and supremum|greatest lower bound (infimum)]] of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit <math>E</math> most tightly and do not overlap.

That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition is tested by taking subsets <math>A</math> of the real numbers using <math>E</math> as an instrument to split <math>A</math> into two partitions: the part of <math>A</math> which intersects with <math>E</math> and the remaining part of <math>A</math> which is not in <math>E</math>: the set difference of <math>A</math> and <math>E</math>.  These partitions of <math>A</math> are subject to the outer measure. If for all possible such subsets <math>A</math> of the real numbers, the partitions of <math>A</math> cut apart by <math>E</math> have outer measures whose sum is the outer measure of <math>A</math>, then the outer Lebesgue measure of <math>E</math> gives its Lebesgue measure. Intuitively, this condition means that the set <math>E</math> must not have some curious properties which causes a discrepancy in the measure of another set when <math>E</math> is used as a "mask" to "clip" that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.)