In mathematical analysis, a null set is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space a null set is a set such that .
Every countable subset of the real numbers that (i.e. finite or countably infinite) is null. For example, the set of natural numbers is countable, having cardinality (aleph-zero or aleph-null), is null. Another example is the set of rational numbers, which is also countable, and hence null.
However, there are some uncountable sets, such as the Cantor set, that are null.
Suppose is a subset of the real line such that
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.
A subset N of has null Lebesgue measure and is considered to be a null set in if and only if:
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 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.
The Borel measure is not complete. One simple construction is to start with the standard Cantor set K, which is closed hence Borel measurable, and which has measure zero, and to find a subset F of K which is not Borel measurable. (Since the Lebesgue measure is complete, this F is of course Lebesgue measurable.)
First, we have to know that every set of positive measure contains a nonmeasurable subset. Let f be the Cantor function, a continuous function which is locally constant on Kc, and monotonically increasing on [0, 1], with f(0) = 0 and f(1) = 1. Obviously, f(Kc) is countable, since it contains one point per component of Kc. Hence f(Kc) has measure zero, so f(K) has measure one. We need a strictly monotonic function, so consider g(x) = f(x) + x. Since g(x) is strictly monotonic and continuous, it is a homeomorphism. Furthermore, g(K) has measure one. Let E ? g(K) be non-measurable, and let F = g-1(E). Because g is injective, we have that F ? K, and so F is a null set. However, if it were Borel measurable, then g(F) would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable; g(F) = (g-1)-1(F) is the preimage of F through the continuous function h = g-1.) Therefore, F is a null, but non-Borel measurable set.
In a separable Banach space (X, +), the group operation moves any subset A ? X to the translates A + x for any x ? X. When there is a probability measure ? on the ?-algebra of Borel subsets of X, such that for all x, ?(A + x) = 0, then A 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 A is not a meagre set then A-1A contains an open neighborhood of the identity element. This property is named for Hugo Steinhaus since it is the conclusion of the Steinhaus theorem.