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# Null set

In measure theory, a null set is a set whose measure is zero, so 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's 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 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 setss 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.