Editing Null set (section)

==Definition for Lebesgue measure==

The [[Lebesgue measure]] is the standard way of assigning a [[length]], [[area]] or [[volume]] to subsets of [[Euclidean space]].

A subset <math>N</math> of the [[real line]] <math>\Reals</math> has null Lebesgue measure and is considered to be a null set (also known as a set of zero-content) in <math>\Reals</math> if and only if:
: [[Given any]] [[positive number]] <math>\varepsilon,</math> [[Existential quantification|there is]] a [[sequence]] <math>I_1, I_2, \ldots</math> of [[interval (mathematics)|intervals]] in <math>\Reals</math> (where interval <math>I_n = (a_n, b_n) \subseteq \Reals</math> has length <math>\operatorname{length}(I_n) =  b_n - a_n</math>) such that <math>N</math> is contained in the union of the <math>I_1, I_2, \ldots</math> and the total length of the union is less than <math>\varepsilon;</math> i.e.,<ref>{{cite book | first=John | last=Franks | date=2009 | title=A (Terse) Introduction to Lebesgue Integration | volume=48 | page=28 | publisher=[[American Mathematical Society]] | isbn=978-0-8218-4862-3 | doi=10.1090/stml/048| series=The Student Mathematical Library }}</ref> <math display="block">
N \subseteq \bigcup_{n=1}^\infty I_n \ ~\textrm{and}~ \ \sum_{n=1}^\infty \operatorname{length}(I_n) < \varepsilon \,.
</math>
(In terminology of [[mathematical analysis]], this definition requires that there be a [[sequence]] of [[open cover]]s of <math>A</math> for which the [[Limit of a sequence|limit]] of the lengths of the covers is zero.)

This condition can be generalised to <math>\Reals^n,</math> using <math>n</math>-[[Cube (geometry)|cube]]s instead of intervals. In fact, the idea can be made to make sense on any [[manifold]], even if there is no Lebesgue measure there.

For instance:
* With respect to <math>\Reals^n,</math> all [[singleton (mathematics)|singleton set]]s are null, and therefore all [[countable set]]s are null. In particular, the set <math>\Q</math> of [[rational number]]s is a null set, despite being [[dense (topology)|dense]] in <math>\Reals.</math>
* The standard construction of the [[Cantor set]] is an example of a null [[uncountable set]] in <math>\Reals;</math> however other constructions are possible which assign the Cantor set any measure whatsoever.
* All the subsets of <math>\Reals^n</math> whose [[dimension]] is smaller than <math>n</math> have null Lebesgue measure in <math>\Reals^n.</math> For instance straight lines or circles are null sets in <math>\Reals^2.</math>
* [[Sard's lemma]]: the set of '''critical values''' of a smooth function has measure zero.

If <math>\lambda</math> is Lebesgue measure for <math>\Reals</math> and π is Lebesgue measure for <math>\Reals^2</math>, then the [[product measure]] <math>\lambda \times \lambda = \pi.</math> In terms of null sets, the following equivalence has been styled a [[Fubini's theorem]]:<ref>{{cite journal | first=Eric K. | last=van Douwen | date=1989 | title=Fubini's theorem for null sets | journal=[[American Mathematical Monthly]] | volume=96 | issue=8 | pages=718–21 | mr=1019152 | jstor=2324722| doi=10.1080/00029890.1989.11972270 }}</ref> 
* For <math>A \subset \Reals^2</math> and <math>A_x = \{y : (x , y) \isin A\},</math> <math display="block">\pi(A) = 0 \iff \lambda \left(\left\{x : \lambda\left(A_x\right) > 0\right\}\right) = 0.</math>