Null set

Summary

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

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 edit

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 edit

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 edit

Let   be a measure space. We have:

  •   (by definition of  ).
  • Any countable union of null sets is itself a null set (by countable subadditivity of  ).
  • Any (measurable) subset of a null set is itself a null set (by monotonicity of  ).

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 edit

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:

  • With respect to   all singleton sets are null, and therefore all countable sets are null. In particular, the set   of rational numbers is a null set, despite being dense in  
  • The standard construction of the Cantor set is an example of a null uncountable set in   however other constructions are possible which assign the Cantor set any measure whatsoever.
  • All the subsets of   whose dimension is smaller than   have null Lebesgue measure in   For instance straight lines or circles are null sets in  
  • Sard's lemma: the set of critical values of a smooth function has measure zero.

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]

  • For   and  
     

Uses edit

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 edit

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 edit

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 edit

  • 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

References edit

  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 edit

  • 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.