Inmathematical analysis, anull set is aLebesgue measurable set of real numbers that hasmeasure zero. This can be characterized as a set that can becovered by acountable union ofintervals of arbitrarily small total length.

The notion of null set should not be confused with theempty set as defined inset theory. Although the empty set hasLebesgue 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 givenmeasure space${\displaystyle M=(X,\mathrm{\Sigma },\mu )}$ a null set is a set${\displaystyle S\in \mathrm{\Sigma }}$ such that${\displaystyle \mu (S)=0.}$

Every finite orcountably infinite subset of thereal numbers⁠${\displaystyle \mathbb{R}}$⁠ is a null set. For example, the set ofnatural numbers⁠${\displaystyle \mathbb{N}}$⁠, the set ofrational numbers⁠${\displaystyle \mathbb{Q}}$⁠ and the set ofalgebraic numbers⁠${\displaystyle \mathbb{A}}$⁠ are all countably infinite and therefore are null sets when considered as subsets of the real numbers.

TheCantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whoseternary expansion can be written using only 0s and 2s (seeCantor'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.

TheLebesgue measure is the standard way of assigning alength,area orvolume to subsets ofEuclidean space.

A subset${\displaystyle N}$ of thereal line${\displaystyle \mathbb{R}}$ has null Lebesgue measure and is considered to be a null set (also known as a set of zero-content) in${\displaystyle \mathbb{R}}$ if and only if:

Given anypositive number${\displaystyle \epsilon ,}$there is asequence${\displaystyle I_1,I_2,\dots }$ ofintervals in${\displaystyle \mathbb{R}}$ (where interval${\displaystyle I_n=(a_n,b_n)\subseteq \mathbb{R}}$ has length${\displaystyle \mathrm{length}(I_n)=b_n-a_n}$) such that${\displaystyle N}$ is contained in the union of the${\displaystyle I_1,I_2,\dots }$ and the total length of the union is less than${\displaystyle \epsilon ;}$ i.e.,^[1]

(In terminology ofmathematical analysis, this definition requires that there be asequence ofopen covers of${\displaystyle A}$ for which thelimit of the lengths of the covers is zero.)

This condition can be generalised to${\displaystyle {\mathbb{R}}^n,}$ using${\displaystyle n}$-cubes instead of intervals. In fact, the idea can be made to make sense on anymanifold, even if there is no Lebesgue measure there.

For instance:

• With respect to${\displaystyle {\mathbb{R}}^n,}$ allsingleton sets are null, and therefore allcountable sets are null. In particular, the set${\displaystyle \mathbb{Q}}$ ofrational numbers is a null set, despite beingdense in${\displaystyle \mathbb{R}.}$
• The standard construction of theCantor set is an example of a nulluncountable set in${\displaystyle \mathbb{R};}$ however other constructions are possible which assign the Cantor set any measure whatsoever.
• All the subsets of${\displaystyle {\mathbb{R}}^n}$ whosedimension is smaller than${\displaystyle n}$ have null Lebesgue measure in${\displaystyle {\mathbb{R}}^n.}$ For instance straight lines or circles are null sets in${\displaystyle {\mathbb{R}}^2.}$
• Sard's lemma: the set ofcritical values of a smooth function has measure zero.

If${\displaystyle \lambda }$ is Lebesgue measure for${\displaystyle \mathbb{R}}$ and π is Lebesgue measure for${\displaystyle {\mathbb{R}}^2}$, then theproduct measure${\displaystyle \lambda \times \lambda =\pi .}$ In terms of null sets, the following equivalence has been styled aFubini's theorem:^[2]

• For${\displaystyle A\subset {\mathbb{R}}^2}$ and${\displaystyle A_x=\{y:(x,y)\in A\},}$$${\displaystyle \pi (A)=0⟺\lambda \left(\left\{x:\lambda \left(A_x\right)>0\right\}\right)=0.}$$

Let${\displaystyle (X,\mathrm{\Sigma },\mu )}$ be ameasure space. We have:

• ${\displaystyle \mu (\varnothing )=0}$ (bydefinition of${\displaystyle \mu }$).
• Anycountableunion of null sets is itself a null set (bycountable subadditivity of${\displaystyle \mu }$).
• Any (measurable) subset of a null set is itself a null set (bymonotonicity of${\displaystyle \mu }$).

Together, these facts show that the null sets of${\displaystyle (X,\mathrm{\Sigma },\mu )}$ form a𝜎-ideal of the𝜎-algebra${\displaystyle \mathrm{\Sigma }}$. Accordingly, null sets may be interpreted asnegligible sets, yielding a measure-theoretic notion of "almost everywhere".

Null sets play a key role in the definition of theLebesgue integral: if functions${\displaystyle f}$ and${\displaystyle g}$ are equal except on a null set, then${\displaystyle f}$ is integrable if and only if${\displaystyle g}$ is, and their integrals are equal. This motivates the formal definition of${\displaystyle L^p}$ 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 iscomplete. 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-completeBorel measure.

The Borel measure is not complete. One simple construction is to start with the standardCantor set${\displaystyle K,}$ which is closed hence Borel measurable, and which has measure zero, and to find a subset${\displaystyle F}$ of${\displaystyle K}$ which is not Borel measurable. (Since the Lebesgue measure is complete, this${\displaystyle F}$ is of course Lebesgue measurable.)

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

In aseparableBanach space${\displaystyle (X,\Vert \cdot \Vert ),}$ addition moves any subset${\displaystyle A\subseteq X}$ to the translates${\displaystyle A+x}$ for any${\displaystyle x\in X.}$ When there is aprobability measureμ on the σ-algebra ofBorel subsets of${\displaystyle X,}$ such that for all${\displaystyle x,}$${\displaystyle \mu (A+x)=0,}$ then${\displaystyle A}$ is aHaar null set.^[3]

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

Some algebraic properties oftopological groups have been related to the size of subsets and Haar null sets.^[4]Haar null sets have been used inPolish groups to show that whenA is not ameagre set then${\displaystyle A^{-1}A}$ contains an open neighborhood of theidentity element.^[5] This property is named forHugo Steinhaus since it is the conclusion of theSteinhaus theorem.

• Capinski, Marek; Kopp, Ekkehard (2005).Measure, Integral and Probability. Springer. p. 16.ISBN 978-1-85233-781-0.
• Jones, Frank (1993).Lebesgue Integration on Euclidean Spaces. Jones & Bartlett. p. 107.ISBN 978-0-86720-203-8.
• Oxtoby, John C. (1971).Measure and Category. Springer-Verlag. p. 3.ISBN 978-0-387-05349-3.

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