View source for Null set

{{Short description|Measurable set whose measure is zero}}
{{For2|the set with no elements|Empty set|the set of zeros of a function|Zero set}}

[[File:Sierpinski triangle.svg|thumb|The [[Sierpiński triangle]] is an example of a null set of points in <math>\mathbb R^2</math>.]]

In [[mathematical analysis]], a '''null set''' is a [[Lebesgue measurable set]] of real numbers that has '''[[Lebesgue measure|measure]] zero'''.  This can be characterized as a set that can be [[Cover (topology)|covered]] by a [[countable]] union of [[Interval (mathematics)|interval]]s of arbitrarily small total length. 

The notion of null set should not be confused with the [[empty set]] as defined in [[set theory]]. Although the empty set has [[Lebesgue measure]] zero, there are also non-empty sets which are null.  For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.

More generally, on a given [[measure space]] <math>M = (X, \Sigma, \mu)</math> a null set is a set <math>S \in \Sigma</math> such that <math>\mu(S) = 0.</math>

==Examples==

Every finite or [[countably infinite]] subset of the [[real numbers]] {{tmath|\R}} is a null set. For example, the set of [[natural numbers]] {{tmath|\N}}, the set of [[rational numbers]] {{tmath|\Q}} and the set of [[algebraic numbers]] {{tmath|\mathbb A}} are all countably infinite and therefore are null sets when considered as subsets of the real numbers.

The [[Cantor set]] is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whose [[Ternary numeral system|ternary]] expansion can be written using only 0s and 2s (see [[Cantor's diagonal argument]]), and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and iteratively removing a third of the previous set, thereby multiplying the length by 2/3 with every step.

==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:

{{block indent| [[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>

==Measure-theoretic properties==

Let <math>(X,\Sigma,\mu)</math> be a [[measure space]]. We have:

* <math>\mu(\varnothing) = 0</math> (by [[Measure_(mathematics)#Definition|definition]] of <math>\mu</math>).
* Any [[countable]] [[union (set theory)|union]] of null sets is itself a null set (by [[Measure_(mathematics)#Countable_subadditivity|countable subadditivity]] of <math>\mu</math>).
* Any (measurable) subset of a null set is itself a null set (by [[Measure_(mathematics)#Monotonicity|monotonicity]] of <math>\mu</math>).

Together, these facts show that the null sets of <math>(X,\Sigma,\mu)</math> form a [[Sigma-ideal|𝜎-ideal]] of the [[sigma-algebra|𝜎-algebra]] <math>\Sigma</math>. Accordingly, null sets may be interpreted as [[negligible set]]s, yielding a measure-theoretic notion of "[[almost everywhere]]".

==Uses==

Null sets play a key role in the definition of the [[Lebesgue integration|Lebesgue integral]]: if functions <math>f</math> and <math>g</math> are equal except on a null set, then <math>f</math> is integrable if and only if <math>g</math> is, and their integrals are equal. This motivates the formal definition of [[Lp space|<math>L^p</math> spaces]] as sets of equivalence classes of functions which differ only on null sets.

A measure in which all subsets of null sets are measurable is ''[[complete measure|complete]]''. Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete [[Borel measure]].

===A subset of the Cantor set which is not Borel measurable===

The Borel measure is not complete. One simple construction is to start with the standard [[Cantor set]] <math>K,</math> which is closed hence Borel measurable, and which has measure zero, and to find a subset <math>F</math> of <math>K</math> which is not Borel measurable. (Since the Lebesgue measure is complete, this <math>F</math> is of course Lebesgue measurable.)

First, we have to know that every set of positive measure contains a nonmeasurable subset. Let <math>f</math> be the [[Cantor function]], a continuous function which is locally constant on <math>K^c,</math> and monotonically increasing on <math>[0, 1],</math> with <math>f(0) = 0</math> and <math>f(1) = 1.</math> Obviously, <math>f(K^c)</math> is countable, since it contains one point per component of <math>K^c.</math> Hence <math>f(K^c)</math> has measure zero, so <math>f(K)</math> has measure one. We need a strictly [[monotonic function]], so consider <math>g(x) = f(x) + x.</math> Since <math>g</math> is strictly monotonic and continuous, it is a [[homeomorphism]]. Furthermore, <math>g(K)</math> has measure one. Let <math>E \subseteq g(K)</math> be non-measurable, and let <math>F = g^{-1}(E).</math> Because <math>g</math> is injective, we have that <math>F \subseteq K,</math> and so <math>F</math> is a null set. However, if it were Borel measurable, then <math>f(F)</math> would also be Borel measurable (here we use the fact that the [[Image (mathematics)|preimage]] of a Borel set by a continuous function is measurable; <math>g(F) = (g^{-1})^{-1}(F)</math> is the preimage of <math>F</math> through the continuous function <math>h = g^{-1}</math>). Therefore <math>F</math> is a null, but non-Borel measurable set.

==Haar null==

In a [[separable space|separable]] [[Banach space]] <math>(X, \|\cdot\|),</math> addition moves any subset <math>A \subseteq X</math> to the translates <math>A + x</math> for any <math>x \in X.</math> When there is a [[probability measure]] {{math|''μ''}} on the σ-algebra of [[Borel subset]]s of <math>X,</math> such that for all <math>x,</math> <math>\mu(A + x) = 0,</math> then <math>A</math> is a '''Haar null set'''.<ref>{{cite journal | first=Eva | last=Matouskova | date=1997 | url=https://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03776-3/S0002-9939-97-03776-3.pdf | title=Convexity and Haar Null Sets | journal=[[Proceedings of the American Mathematical Society]] | volume=125 | issue=6 | pages=1793–1799 | jstor=2162223| doi=10.1090/S0002-9939-97-03776-3 | doi-access=free }}</ref>

The term refers to the null invariance of the measures of translates, associating it with the complete invariance found with [[Haar measure]].

Some algebraic properties of [[topological group]]s have been related to the size of subsets and Haar null sets.<ref>{{cite journal | first=S. | last=Solecki | date=2005 | title=Sizes of subsets of groups and Haar null sets | journal=Geometric and Functional Analysis | volume=15 | pages=246–73 | mr=2140632 | doi=10.1007/s00039-005-0505-z| citeseerx=10.1.1.133.7074 | s2cid=11511821 }}</ref>
Haar null sets have been used in [[Polish group]]s to show that when {{mvar|A}} is not a [[meagre set]] then <math>A^{-1} A</math> contains an open neighborhood of the [[identity element]].<ref>{{cite journal | first=Pandelis | last=Dodos | date=2009 | title=The Steinhaus property and Haar-null sets | journal=[[Bulletin of the London Mathematical Society]] | volume=41 | issue=2 | pages=377–44 | mr=4296513| bibcode=2010arXiv1006.2675D | arxiv=1006.2675 | doi=10.1112/blms/bdp014 | s2cid=119174196 }}</ref> This property is named for [[Hugo Steinhaus]] since it is the conclusion of the [[Steinhaus theorem]].

==References==

{{reflist}}

==Further reading==

* {{cite book|last1=Capinski|first1=Marek|last2=Kopp|first2=Ekkehard|date=2005|title=Measure, Integral and Probability|page=16|publisher=Springer|isbn=978-1-85233-781-0}}
* {{cite book|last=Jones|first=Frank|date=1993|title=Lebesgue Integration on Euclidean Spaces|page=107|publisher=Jones & Bartlett|isbn=978-0-86720-203-8}}
* {{cite book|last=Oxtoby|first=John C.|date=1971|title=Measure and Category|page=3|publisher=Springer-Verlag|isbn=978-0-387-05349-3}}

{{Measure theory}}

[[Category:Measure theory]]
[[Category:Set theory]]