# Null set

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Measurable set whose measure is zero

For the set with no elements, see [Empty set](/source/Empty_set). For the set of zeros of a function, see [Zero set](/source/Zero_set).

The [Sierpiński triangle](/source/Sierpi%C5%84ski_triangle) is an example of a null set of points in

            R

            2

    {\displaystyle \mathbb {R} ^{2}}

.

In [mathematical analysis](/source/Mathematical_analysis), a **null set** is a [Lebesgue measurable set](/source/Lebesgue_measurable_set) of real numbers that has **[measure](/source/Lebesgue_measure) zero**. This can be characterized as a set that can be [covered](/source/Cover_(topology)) by a [countable](/source/Countable) union of [intervals](/source/Interval_(mathematics)) of arbitrarily small total length.

A null set is not to be confused with the [empty set](/source/Empty_set) as defined in [set theory](/source/Set_theory). Although the empty set has [Lebesgue measure](/source/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](/source/Measure_space) M = ( X , Σ , μ ) {\displaystyle M=(X,\Sigma ,\mu )} a null set is a set S ∈ Σ {\displaystyle S\in \Sigma } such that μ ( S ) = 0. {\displaystyle \mu (S)=0.}

## Examples

Every finite or [countably infinite](/source/Countably_infinite) subset of the [real numbers](/source/Real_numbers) ⁠ R {\displaystyle \mathbb {R} } ⁠ is a null set. For example, the set of [natural numbers](/source/Natural_numbers) ⁠ N {\displaystyle \mathbb {N} } ⁠, the set of [rational numbers](/source/Rational_numbers) ⁠ Q {\displaystyle \mathbb {Q} } ⁠ and the set of [algebraic numbers](/source/Algebraic_numbers) ⁠ A {\displaystyle \mathbb {A} } ⁠ are all countably infinite and therefore are null sets when considered as subsets of the real numbers.

The [Cantor set](/source/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](/source/Ternary_numeral_system) expansion can be written using only 0s and 2s (see [Cantor's diagonal argument](/source/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.

The set of [Liouville numbers](/source/Liouville_number) is another example of an uncountable null set.

## Definition for Lebesgue measure

The [Lebesgue measure](/source/Lebesgue_measure) is the standard way of assigning a [length](/source/Length), [area](/source/Area) or [volume](/source/Volume) to subsets of [Euclidean space](/source/Euclidean_space).

A subset N {\displaystyle N} of the [real line](/source/Real_line) R {\displaystyle \mathbb {R} } has null Lebesgue measure and is considered to be a null set (also known as a set of zero-content) in R {\displaystyle \mathbb {R} } if and only if:

 [Given any](/source/Given_any) [positive number](/source/Positive_number)

        ε
        ,

    {\displaystyle \varepsilon ,}

 [there is](/source/Existential_quantification) a [sequence](/source/Sequence)

          I

            1

        ,

          I

            2

        ,
        …

    {\displaystyle I_{1},I_{2},\ldots }

 of [intervals](/source/Interval_(mathematics)) in

          R

    {\displaystyle \mathbb {R} }

 (where interval

          I

            n

        =
        (

          a

            n

        ,

          b

            n

        )
        ⊆

          R

    {\displaystyle I_{n}=(a_{n},b_{n})\subseteq \mathbb {R} }

 has length

        length
        ⁡
        (

          I

            n

        )
        =

          b

            n

        −

          a

            n

    {\displaystyle \operatorname {length} (I_{n})=b_{n}-a_{n}}

) such that

        N

    {\displaystyle N}

 is contained in the union of the

          I

            1

        ,

          I

            2

        ,
        …

    {\displaystyle I_{1},I_{2},\ldots }

 and the total length of the union is less than

        ε
        ;

    {\displaystyle \varepsilon ;}

 i.e.,[1]

        N
        ⊆

          ⋃

            n
            =
            1

            ∞

          I

            n

         
         

            and

         
         

          ∑

            n
            =
            1

            ∞

        length
        ⁡
        (

          I

            n

        )
        <
        ε

        .

    {\displaystyle N\subseteq \bigcup _{n=1}^{\infty }I_{n}\ ~{\textrm {and}}~\ \sum _{n=1}^{\infty }\operatorname {length} (I_{n})<\varepsilon \,.}

(In terminology of [mathematical analysis](/source/Mathematical_analysis), this definition requires that there be a [sequence](/source/Sequence) of [open covers](/source/Open_cover) of A {\displaystyle A} for which the [limit](/source/Limit_of_a_sequence) of the lengths of the covers is zero.)

This condition can be generalised to R n , {\displaystyle \mathbb {R} ^{n},} using n {\displaystyle n} -[cubes](/source/Cube_(geometry)) instead of intervals. In fact, the idea can be made to make sense on any [manifold](/source/Manifold), even if there is no Lebesgue measure there.

For instance:

- With respect to R n , {\displaystyle \mathbb {R} ^{n},} all [singleton sets](/source/Singleton_(mathematics)) are null, and therefore all [countable sets](/source/Countable_set) are null. In particular, the set Q {\displaystyle \mathbb {Q} } of [rational numbers](/source/Rational_number) is a null set, despite being [dense](/source/Dense_(topology)) in R . {\displaystyle \mathbb {R} .}

- The standard construction of the [Cantor set](/source/Cantor_set) is an example of a null [uncountable set](/source/Uncountable_set) in R ; {\displaystyle \mathbb {R} ;} however other constructions are possible which assign the Cantor set any measure whatsoever.

- All the subsets of R n {\displaystyle \mathbb {R} ^{n}} whose [dimension](/source/Dimension) is smaller than n {\displaystyle n} have null Lebesgue measure in R n . {\displaystyle \mathbb {R} ^{n}.} For instance straight lines or circles are null sets in R 2 . {\displaystyle \mathbb {R} ^{2}.}

- [Sard's lemma](/source/Sard's_lemma): the set of **critical values** of a smooth function has measure zero.

If λ {\displaystyle \lambda } is Lebesgue measure for R {\displaystyle \mathbb {R} } and π is Lebesgue measure for R 2 {\displaystyle \mathbb {R} ^{2}} , then the [product measure](/source/Product_measure) λ × λ = π . {\displaystyle \lambda \times \lambda =\pi .} In terms of null sets, the following equivalence has been styled a [Fubini's theorem](/source/Fubini's_theorem):[2]

- For A ⊂ R 2 {\displaystyle A\subset \mathbb {R} ^{2}} and A x = { y : ( x , y ) ∈ A } , {\displaystyle A_{x}=\{y:(x,y)\in A\},} π ( A ) = 0 ⟺ λ ( { x : λ ( A x ) > 0 } ) = 0. {\displaystyle \pi (A)=0\iff \lambda \left(\left\{x:\lambda \left(A_{x}\right)>0\right\}\right)=0.}

## Measure-theoretic properties

Let ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} be a [measure space](/source/Measure_space). We have:

- μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0} (by [definition](/source/Measure_(mathematics)#Definition) of μ {\displaystyle \mu } ).

- Any [countable](/source/Countable) [union](/source/Union_(set_theory)) of null sets is itself a null set (by [countable subadditivity](/source/Measure_(mathematics)#Countable_subadditivity) of μ {\displaystyle \mu } ).

- Any (measurable) subset of a null set is itself a null set (by [monotonicity](/source/Measure_(mathematics)#Monotonicity) of μ {\displaystyle \mu } ).

Together, these facts show that the null sets of ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} form a [𝜎-ideal](/source/Sigma-ideal) of the [𝜎-algebra](/source/Sigma-algebra) Σ {\displaystyle \Sigma } . Accordingly, null sets may be interpreted as [negligible sets](/source/Negligible_set), yielding a measure-theoretic notion of "[almost everywhere](/source/Almost_everywhere)".

## Uses

Null sets play a key role in the definition of the [Lebesgue integral](/source/Lebesgue_integration): if functions f {\displaystyle f} and g {\displaystyle g} are equal except on a null set, then f {\displaystyle f} is integrable if and only if g {\displaystyle g} is, and their integrals are equal. This motivates the formal definition of [L p {\displaystyle L^{p}} spaces](/source/Lp_space) 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](/source/Complete_measure)*. 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](/source/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](/source/Cantor_set) K , {\displaystyle K,} which is closed hence Borel measurable, and which has measure zero, and to find a subset F {\displaystyle F} of K {\displaystyle K} which is not Borel measurable. (Since the Lebesgue measure is complete, this F {\displaystyle F} is of course Lebesgue measurable.)

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

## Haar null

In a [separable](/source/Separable_space) [Banach space](/source/Banach_space) ( X , ‖ ⋅ ‖ ) , {\displaystyle (X,\|\cdot \|),} addition moves any subset A ⊆ X {\displaystyle A\subseteq X} to the translates A + x {\displaystyle A+x} for any x ∈ X . {\displaystyle x\in X.} When there is a [probability measure](/source/Probability_measure) *μ* on the σ-algebra of [Borel subsets](/source/Borel_subset) of X , {\displaystyle X,} such that for all x , {\displaystyle x,} μ ( A + x ) = 0 , {\displaystyle \mu (A+x)=0,} then A {\displaystyle A} is a **Haar null set**.[3]

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

Some algebraic properties of [topological groups](/source/Topological_group) have been related to the size of subsets and Haar null sets.[4] Haar null sets have been used in [Polish groups](/source/Polish_group) to show that when A is not a [meagre set](/source/Meagre_set) then A − 1 A {\displaystyle A^{-1}A} contains an open neighborhood of the [identity element](/source/Identity_element).[5] This property is named for [Hugo Steinhaus](/source/Hugo_Steinhaus) since it is the conclusion of the [Steinhaus theorem](/source/Steinhaus_theorem).

## References

1. **[^](#cite_ref-1)** Franks, John (2009). *A (Terse) Introduction to Lebesgue Integration*. The Student Mathematical Library. Vol. 48. [American Mathematical Society](/source/American_Mathematical_Society). p. 28. [doi](/source/Doi_(identifier)):[10.1090/stml/048](https://doi.org/10.1090%2Fstml%2F048). [ISBN](/source/ISBN_(identifier)) [978-0-8218-4862-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-4862-3).

1. **[^](#cite_ref-2)** van Douwen, Eric K. (1989). "Fubini's theorem for null sets". *[American Mathematical Monthly](/source/American_Mathematical_Monthly)*. **96** (8): 718–21. [doi](/source/Doi_(identifier)):[10.1080/00029890.1989.11972270](https://doi.org/10.1080%2F00029890.1989.11972270). [JSTOR](/source/JSTOR_(identifier)) [2324722](https://www.jstor.org/stable/2324722). [MR](/source/MR_(identifier)) [1019152](https://mathscinet.ams.org/mathscinet-getitem?mr=1019152).

1. **[^](#cite_ref-3)** Matouskova, Eva (1997). ["Convexity and Haar Null Sets"](https://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03776-3/S0002-9939-97-03776-3.pdf) (PDF). *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **125** (6): 1793–1799. [doi](/source/Doi_(identifier)):[10.1090/S0002-9939-97-03776-3](https://doi.org/10.1090%2FS0002-9939-97-03776-3). [JSTOR](/source/JSTOR_(identifier)) [2162223](https://www.jstor.org/stable/2162223).

1. **[^](#cite_ref-4)** Solecki, S. (2005). "Sizes of subsets of groups and Haar null sets". *Geometric and Functional Analysis*. **15**: 246–73. [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.133.7074](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.133.7074). [doi](/source/Doi_(identifier)):[10.1007/s00039-005-0505-z](https://doi.org/10.1007%2Fs00039-005-0505-z). [MR](/source/MR_(identifier)) [2140632](https://mathscinet.ams.org/mathscinet-getitem?mr=2140632). [S2CID](/source/S2CID_(identifier)) [11511821](https://api.semanticscholar.org/CorpusID:11511821).

1. **[^](#cite_ref-5)** Dodos, Pandelis (2009). "The Steinhaus property and Haar-null sets". *[Bulletin of the London Mathematical Society](/source/Bulletin_of_the_London_Mathematical_Society)*. **41** (2): 377–44. [arXiv](/source/ArXiv_(identifier)):[1006.2675](https://arxiv.org/abs/1006.2675). [Bibcode](/source/Bibcode_(identifier)):[2010arXiv1006.2675D](https://ui.adsabs.harvard.edu/abs/2010arXiv1006.2675D). [doi](/source/Doi_(identifier)):[10.1112/blms/bdp014](https://doi.org/10.1112%2Fblms%2Fbdp014). [MR](/source/MR_(identifier)) [4296513](https://mathscinet.ams.org/mathscinet-getitem?mr=4296513). [S2CID](/source/S2CID_(identifier)) [119174196](https://api.semanticscholar.org/CorpusID:119174196).

## Further reading

- Capinski, Marek; Kopp, Ekkehard (2005). *Measure, Integral and Probability*. Springer. p. 16. [ISBN](/source/ISBN_(identifier)) [978-1-85233-781-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85233-781-0).

- Jones, Frank (1993). *Lebesgue Integration on Euclidean Spaces*. Jones & Bartlett. p. 107. [ISBN](/source/ISBN_(identifier)) [978-0-86720-203-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-86720-203-8).

- Oxtoby, John C. (1971). *Measure and Category*. Springer-Verlag. p. 3. [ISBN](/source/ISBN_(identifier)) [978-0-387-05349-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-05349-3).

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