Null set

In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero.

This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.

For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.

More generally, on a given measure space

Every finite or countably infinite subset of the real numbers ⁠

⁠, the set of rational numbers ⁠

⁠ are all countably infinite and therefore are null sets when considered as subsets of the real numbers.

It is uncountable because it contains all real numbers between 0 and 1 whose ternary form decimal expansion can be written using only 0’s and 2’s, and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and multiplying the length by 2/3 continuously.

In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of

for which the limit of the lengths of the covers is zero.

We have: Together, these facts show that the null sets of

form a 𝜎-ideal of the 𝜎-algebra

Accordingly, null sets may be interpreted as negligible sets, yielding a measure-theoretic notion of "almost everywhere".

The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space.

In fact, the idea can be made to make sense on any manifold, even if there is no Lebesgue measure there.

In terms of null sets, the following equivalence has been styled a Fubini's theorem:[2] Null sets play a key role in the definition of the Lebesgue integral: if functions

spaces as sets of equivalence classes of functions which differ only on null sets.

Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero.

One simple construction is to start with the standard Cantor set

(Since the Lebesgue measure is complete, this

First, we have to know that every set of positive measure contains a nonmeasurable subset.

We need a strictly monotonic function, so consider

is strictly monotonic and continuous, it is a homeomorphism.

would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable;

is a null, but non-Borel measurable set.

When there is a probability measure μ on the σ-algebra of Borel subsets of

[3] 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.

[4] Haar null sets have been used in Polish groups to show that when A is not a meagre set then

contains an open neighborhood of the identity element.