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 {In} of intervals such that N is contained in the union of the In and the total length of the In 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.