Editing Lebesgue measure (section)

== Examples ==

* Any closed [[Interval (mathematics)|interval]] <math display="inline">[a, b]</math> of [[real number]]s is Lebesgue-measurable, and its Lebesgue measure is the length <math display="inline">b - a</math>. The [[open interval]] <math display="inline">(a, b)</math> has the same measure, since the [[set difference|difference]] between the two sets consists only of the end points <math>a</math> and <math>b</math>, which each have [[measure zero]].
* Any [[Cartesian product]] of intervals <math display="inline">[a, b]</math> and <math display="inline">[c, d]</math> is Lebesgue-measurable, and its Lebesgue measure is <math display="inline">(b - a)(c-d)</math>, the area of the corresponding [[rectangle]].
* Moreover, every [[Borel set]] is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.<ref>{{cite web | url=https://math.stackexchange.com/q/556756 | title=What sets are Lebesgue-measurable? | publisher=math stack exchange | access-date=26 September 2015 | author=Asaf Karagila}}</ref><ref>{{cite web | url=https://math.stackexchange.com/q/142385 | title=Is there a sigma-algebra on R strictly between the Borel and Lebesgue algebras? | publisher=math stack exchange | access-date=26 September 2015 | author=Asaf Karagila}}</ref>
* Any [[countable]] set of real numbers has Lebesgue measure {{Math|0}}. In particular, the Lebesgue measure of the set of [[algebraic numbers]] is {{Math|0}}, even though the set is [[Dense set|dense]] in <math>\mathbb{R}</math>.
* The [[Cantor set]] and the set of [[Liouville number]]s are examples of [[uncountable set]]s that have Lebesgue measure {{Math|0}}.
* If the [[axiom of determinacy]] holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the [[axiom of choice]].
* [[Vitali set]]s are examples of sets that are [[non-measurable set|not measurable]] with respect to the Lebesgue measure.  Their existence relies on the [[axiom of choice]].
* [[Osgood curve]]s are simple plane [[curve]]s with [[positive number|positive]] Lebesgue measure<ref>{{cite journal|last=Osgood|first=William F.|date=January 1903|title=A Jordan Curve of Positive Area|journal=Transactions of the American Mathematical Society|publisher=American Mathematical Society|volume=4|issue=1|pages=107–112|doi=10.2307/1986455|issn=0002-9947|jstor=1986455|author-link1=William Fogg Osgood|doi-access=free}}<!--|access-date=2008-06-04--></ref> (it can be obtained by small variation of the [[Peano curve]] construction). The [[dragon curve]] is another unusual example.
* Any line in <math>\mathbb{R}^n</math>, for <math>n \geq 2</math>, has a zero Lebesgue measure. In general, every proper [[hyperplane]] has a zero Lebesgue measure in its [[ambient space]].
* The [[volume of an n-ball|volume of an ''{{Math|n}}''-ball]] can be calculated in terms of Euler's gamma function.