Fdtxjr

In mathematical analysis, a null set N⊂Rdisplaystyle Nsubset mathbb R is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space M=(X,Σ,μ)displaystyle M=(X,Sigma ,mu ) a null set is a set S⊂Xdisplaystyle Ssubset X such that μ(S)=0displaystyle mu (S)=0.

Definition

Suppose Adisplaystyle A is a subset of the real line Rdisplaystyle mathbb R such that

∀ε>0,∃Un(A⊂⋃n=1∞Un ∧ ∑n=1∞|Un|<ε)displaystyle forall varepsilon >0,exists leftU_nrightleft(Asubset bigcup _n=1^infty U_n land sum _n=1^infty left

where the U_n are intervals and |U| is the length of U, then A is a null set.^[1] Also known as a set of zero-content.

In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of A for which the limit of the lengths of the covers is zero.

Null sets include all finite sets, all countable sets, and even some uncountable sets such as the Cantor set.

Properties

The empty set is always a null set. More generally, any countable union of null sets is null. Any measurable subset of a null set is itself a null set. Together, these facts show that the m-null sets of X form a sigma-ideal on X. Similarly, the measurable m-null sets form a sigma-ideal of the sigma-algebra of measurable sets. Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhere.

Lebesgue measure

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

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

Given any positive number ε, there is a sequence I_n of intervals in Rdisplaystyle mathbb R such that N is contained in the union of the I_n and the total length of the union is less than ε.

This condition can be generalised to Rndisplaystyle mathbb R ^n, using n-cubes instead of intervals. In fact, the idea can be made to make sense on any Riemannian manifold, even if there is no Lebesgue measure there.

For instance:

• With respect to Rndisplaystyle mathbb R ^n, all 1-point sets are null, and therefore all countable sets are null. In particular, the set Q of rational numbers is a null set, despite being dense in Rdisplaystyle mathbb R .


• The standard construction of the Cantor set is an example of a null uncountable set in Rdisplaystyle mathbb R ; however other constructions are possible which assign the Cantor set any measure whatsoever.


• All the subsets of Rndisplaystyle mathbb R ^n whose dimension is smaller than n have null Lebesgue measure in Rndisplaystyle mathbb R ^n. For instance straight lines or circles are null sets in R2displaystyle mathbb R ^2.


• 
  Sard's lemma: the set of critical values of a smooth function has measure zero.

If λ is Lebesgue measure for Rdisplaystyle mathbb R and π is Lebesgue measure for R2displaystyle mathbb R ^2, then the product measure λ×λ=πdisplaystyle lambda times lambda =pi . In terms of null sets, the following equivalence has been styled a Fubini's theorem:^[2]

• For A⊂R2displaystyle Asubset mathbb R ^2 and Ax=y:(x,y)∈A,displaystyle A_x=y:(x,y)in A,
  

  (π(A)=0)≡λ(x:λ(Ax)>0)=0.displaystyle (pi (A)=0)equiv lambda left(leftx:lambda left(A_xright)>0rightright)=0.

Uses

Null sets play a key role in the definition of the Lebesgue 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.

A measure in which all subsets of null sets are measurable is 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 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 the Cantor 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 strictly monotonic function, so consider g(x) = f(x) + x. Since g(x) is strictly monotonic and continuous, it is a homeomorphism. Furthermore, g(K) has measure one. Let E ⊂ g(K) be non-measurable, and let F = g⁻¹(E). Because g is injective, we have that F ⊂ K, and so F is a null set. However, if it were Borel measurable, then g(F) would also be Borel measurable (here we use the fact that the preimage of 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 a separable Banach space (X, +), the group operation moves any subset A ⊂ X to the translates A + x for any x ∈ X. When there is a probability measure μ on the σ-algebra of Borel subsets of X, such that for all x, μ(A + x) = 0, then 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.

Some algebraic properties of topological groups have been related to the size of subsets and Haar null sets.^[4]
Haar null sets have been used in Polish groups to show that when A is not a meagre set then A^–1A contains an open neighborhood of the identity element.^[5] This property is named for Hugo Steinhaus since it is the conclusion of the Steinhaus theorem.

See also

• Cantor function


• Measure (mathematics)


• Empty set

References

Further reading

• 
  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.