In measure theory, a null set is a set that it is negligible for the purposes of the measure in question. Which sets are null will depend on the measure considered. Thus one may speak of m-null sets for a given measure m.

The term "null set" is sometimes also used to refer to the empty set; see that article. Alternatively, it may be used for any notion of negligible set; see that article. Wikipedia uses the term "null set" only in the measure theoretic sense.

Definition

Let X be a measurable space, let m be a measure on X, and let N be a measurable set in X. If m is a positive measure, then N is null if its measure m(N) is zero. If m is not a positive measure, then N is m-null if N is |m|-null, where |m| is the total variation of m; this is stronger than simply saying that m(N) = 0.

A nonmeasurable set is considered null if it is a subset of a null measurable set. Some references require a null set to be measurable; however, subsets of null sets are still negligible for measure-theoretic purposes.

When talking about null sets in Euclidean n-space Rⁿ, it is usually understood that the measure being used is Lebesgue measure.

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.

In Lebesgue measure

For Lebesgue measure on Rⁿ, 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 R. The Cantor set is an example of an uncountable null set in R.

More generally, a subset N of R is null if and only if:

Given any positive number e, there is a sequence {I_n} of intervals such that N is contained in the union of the I_n and the total length of the I_n is less than e.

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

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 null sets are measurable is complete. Any non-complete measure can be completed to form a complete measure, by assuming that null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it's defined as the completion of a non-complete Borel measure.