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.

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, 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

The empty set is always a null set. More generally, any countable union of null sets is null. Any subset of a null set is itself a null set. Together, these facts show that the -null sets of  form a sigma-ideal on  Similarly, the measurable -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  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: 
 For  and

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  on the σ-algebra of Borel subsets of  such that for all   then  is a Haar null set.

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.
Haar null sets have been used in Polish groups to show that when  is not a meagre set then  contains an open neighborhood of the identity element. This property is named for Hugo Steinhaus since it is the conclusion of the Steinhaus theorem.

See also

References

Further reading

 
 
 

Measure theory
Set theory