The Sierpiński triangle is an example of a null set of points in .

In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals 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 a null set is a set such that

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.

The Cantor set is an example of an uncountable null set.[further explanation needed]

Definition

Suppose is a subset of the real line such that for every there exists a sequence of open intervals (where interval has length ) such that

then 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 for which the limit of the lengths of the covers is zero.

Properties

Let be a measure space. We have:

Together, these facts show that the null sets of form a 𝜎-ideal of the 𝜎-algebra . Accordingly, null sets may be interpreted as negligible sets, yielding a measure-theoretic 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 of has null Lebesgue measure and is considered to be a null set in if and only if:

Given any positive number there is a sequence of intervals in such that is contained in the union of the and the total length of the union is less than

This condition can be generalised to using -cubes 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:

If is Lebesgue measure for and π is Lebesgue measure for , then the product measure In terms of null sets, the following equivalence has been styled a Fubini's theorem:[2]

Uses

Null sets play a key role in the definition of the Lebesgue integral: if functions and are equal except on a null set, then is integrable if and only if is, and their integrals are equal. This motivates the formal definition of 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. 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 which is closed hence Borel measurable, and which has measure zero, and to find a subset of which is not Borel measurable. (Since the Lebesgue measure is complete, this is of course Lebesgue measurable.)

First, we have to know that every set of positive measure contains a nonmeasurable subset. Let be the Cantor function, a continuous function which is locally constant on and monotonically increasing on with and Obviously, is countable, since it contains one point per component of Hence has measure zero, so has measure one. We need a strictly monotonic function, so consider Since is strictly monotonic and continuous, it is a homeomorphism. Furthermore, has measure one. Let be non-measurable, and let Because is injective, we have that and so is a null set. However, if it were Borel measurable, then would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable; is the preimage of through the continuous function ) Therefore, is a null, but non-Borel measurable set.

Haar null

In a separable Banach space the group operation moves any subset to the translates for any When there is a probability measure μ on the σ-algebra of Borel subsets of such that for all then 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 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

References

  1. ^ Franks, John (2009). A (Terse) Introduction to Lebesgue Integration. The Student Mathematical Library. Vol. 48. American Mathematical Society. p. 28. doi:10.1090/stml/048. ISBN 978-0-8218-4862-3.
  2. ^ van Douwen, Eric K. (1989). "Fubini's theorem for null sets". American Mathematical Monthly. 96 (8): 718–21. doi:10.1080/00029890.1989.11972270. JSTOR 2324722. MR 1019152.
  3. ^ Matouskova, Eva (1997). "Convexity and Haar Null Sets" (PDF). Proceedings of the American Mathematical Society. 125 (6): 1793–1799. doi:10.1090/S0002-9939-97-03776-3. JSTOR 2162223.
  4. ^ Solecki, S. (2005). "Sizes of subsets of groups and Haar null sets". Geometric and Functional Analysis. 15: 246–73. CiteSeerX 10.1.1.133.7074. doi:10.1007/s00039-005-0505-z. MR 2140632. S2CID 11511821.
  5. ^ Dodos, Pandelis (2009). "The Steinhaus property and Haar-null sets". Bulletin of the London Mathematical Society. 41 (2): 377–44. arXiv:1006.2675. Bibcode:2010arXiv1006.2675D. doi:10.1112/blms/bdp014. MR 4296513. S2CID 119174196.

Further reading