Null set

In measure theory, a null set is a set of measure 0. Null sets should not be confused with the empty set, although the empty set is always null. When talking about null sets in Rⁿ (Euclidean n-space), it is usually understood that the measure being used is the Lebesgue measure on Rⁿ.

Any countable union of null sets is itself a null set. For Lebesgue measure on Rⁿ, all 1-point sets are null, and therefore all countable sets are null.

Any measurable subset of a null set is itself a null set. A measure in which all subsets of null sets are measurable, and therefore null, is said to be a complete measure. The Lebesgue measure on Rⁿ is an example of a complete measure.

A subset N of R is null if and only if it satisfies the following condition:

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

As mentioned above, all countable subsets of R are null. In particular, Q (the set 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.

Null sets play a key role in the definition of the Lebesgue integral: if functions f and g agree almost everywhere (that is, on the complement of a null set) then f is integrable if and only if g is, and their integrals are equal.