Null set

For the set with no elements, seeEmpty set.

For the set of zeros of a function, seeZero set.

TheSierpiński triangleis an example of a null set of points in$\mathbb {R} ^{2}$.

Inmathematical analysis,anull setis aLebesgue measurable setof real numbers that hasmeasurezero.This can be characterized as a set that can becoveredby acountableunion ofintervalsof arbitrarily small total length.

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

Examples

Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers.

TheCantor setis an example of an uncountable null set.

Definition

Suppose$A$is a subset of thereal line$\mathbb {R}$such that for every$\varepsilon >0,$there exists a sequence$U_{1},U_{2},\ldots$of openintervals(where interval$U_{n}=(a_{n},b_{n})\subseteq \mathbb {R}$has length$\operatorname {length} (U_{n})=b_{n}-a_{n}$such that $$A\subseteq \bigcup _{n=1}^{\infty }U_{n}\ ~{\textrm {and}}~\ \sum _{n=1}^{\infty }\operatorname {length} (U_{n})<\varepsilon \,,$$ then$A$is a null set,also known as a set of zero-content.

In terminology ofmathematical analysis,this definition requires that there be asequenceofopen coversof$A$for which thelimitof the lengths of the covers is zero.

Properties

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

• $\mu (\varnothing )=0$(bydefinitionof$\mu$).
• Anycountableunionof null sets is itself a null set (bycountable subadditivityof$\mu$).
• Any (measurable) subset of a null set is itself a null set (bymonotonicityof$\mu$).

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

Lebesgue measure

TheLebesgue measureis the standard way of assigning alength,areaorvolumeto subsets ofEuclidean space.

A subset$N$of$\mathbb {R}$has null Lebesgue measure and is considered to be a null set in$\mathbb {R}$if and only if:

Given anypositive number$\varepsilon,$there isasequence$I_{1},I_{2},\ldots$ofintervalsin$\mathbb {R}$such that$N$is contained in the union of the$I_{1},I_{2},\ldots$and the total length of the union is less than$\varepsilon.$

This condition can be generalised to$\mathbb {R} ^{n},$using$n$-cubesinstead 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$\mathbb {R} ^{n},$allsingleton setsare null, and therefore allcountable setsare null. In particular, the set$\mathbb {Q}$ofrational numbersis a null set, despite beingdensein$\mathbb {R}.$
• The standard construction of theCantor setis an example of a nulluncountable setin$\mathbb {R};$however other constructions are possible which assign the Cantor set any measure whatsoever.
• All the subsets of$\mathbb {R} ^{n}$whosedimensionis smaller than$n$have null Lebesgue measure in$\mathbb {R} ^{n}.$For instance straight lines or circles are null sets in$\mathbb {R} ^{2}.$
• Sard's lemma:the set ofcritical valuesof a smooth function has measure zero.

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

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

Uses

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

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 standardCantor set$K,$which is closed hence Borel measurable, and which has measure zero, and to find a subset$F$of$K$which is not Borel measurable. (Since the Lebesgue measure is complete, this$F$is of course Lebesgue measurable.)

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

Haar null

In aseparableBanach space$(X,+),$the group operation moves any subset$A\subseteq X$to the translates$A+x$for any$x\in X.$When there is aprobability measureμon the σ-algebra ofBorel subsetsof$X,$such that for all$x,$$\mu (A+x)=0,$then$A$is aHaar null set.

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 groupshave been related to the size of subsets and Haar null sets. Haar null sets have been used inPolish groupsto show that whenAis not ameagre setthen$A^{-1}A$contains an open neighborhood of theidentity element.This property is named forHugo Steinhaussince it is the conclusion of theSteinhaus theorem.

See also

• Cantor function– Continuous function that is not absolutely continuous
• Empty set– Mathematical set containing no elements
• Measure (mathematics)– Generalization of mass, length, area and volume
• Nothing– Complete absence of anything; the opposite of everything

Further reading

• Capinski, Marek; Kopp, Ekkehard (2005).Measure, Integral and Probability.Springer. p. 16.ISBN978-1-85233-781-0.
• Jones, Frank (1993).Lebesgue Integration on Euclidean Spaces.Jones & Bartlett. p. 107.ISBN978-0-86720-203-8.
• Oxtoby, John C. (1971).Measure and Category.Springer-Verlag. p. 3.ISBN978-0-387-05349-3.