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 the set with no elements, see Empty set.

The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. 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 $M=(X,\Sigma ,\mu )$ a null set is a set $S\in \Sigma$ such that $\mu (S)=0.$

Examples[edit]

Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers.

The Cantor set is an example of an uncountable null set.

Definition[edit]

Suppose $A$ is a subset of the real line $\mathbb {R}$ such that for every $\varepsilon >0,$ there exists a sequence $U_{1},U_{2},\ldots$ of open intervals (where interval $U_{n}=(a_{n},b_{n})\subseteq \mathbb {R}$ has length $\operatorname {length} (U_{n})=b_{n}-a_{n}$ such that $A\subseteq \bigcup _{n=1}^{\infty }U_{n}\ ~{\textrm {and}}~\ \sum _{n=1}^{\infty }\operatorname {length} (U_{n})<\varepsilon \,,$ then $A$ is a null set,^[1] also known as a set of zero-content.

In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of $A$ for which the limit of the lengths of the covers is zero.

$\mu (\varnothing )=0$ (by of $\mu$ ).

definition

Any union of null sets is itself a null set (by countable subadditivity of $\mu$ ).

countable

Any (measurable) subset of a null set is itself a null set (by of $\mu$ ).

monotonicity

Let $(X,\Sigma ,\mu )$ be a measure space. We have:

Together, these facts show that the null sets of $(X,\Sigma ,\mu )$ form a 𝜎-ideal of the 𝜎-algebra $\Sigma$ . Accordingly, null sets may be interpreted as negligible sets, yielding a measure-theoretic notion of "almost everywhere".

With respect to $\mathbb {R} ^{n},$ all are null, and therefore all countable sets are null. In particular, the set $\mathbb {Q}$ of rational numbers is a null set, despite being dense in $\mathbb {R} .$

singleton sets

The standard construction of the is an example of a null uncountable set in $\mathbb {R} ;$ however other constructions are possible which assign the Cantor set any measure whatsoever.

Cantor set

All the subsets of $\mathbb {R} ^{n}$ whose is smaller than $n$ have null Lebesgue measure in $\mathbb {R} ^{n}.$ For instance straight lines or circles are null sets in $\mathbb {R} ^{2}.$

dimension

: the set of critical values of a smooth function has measure zero.

Sard's lemma

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

A subset $N$ of $\mathbb {R}$ has null Lebesgue measure and is considered to be a null set in $\mathbb {R}$ if and only if:

This condition can be generalised to $\mathbb {R} ^{n},$ using $n$ -cubes instead of intervals. In fact, the idea can be made to make sense on any manifold, even if there is no Lebesgue measure there.

For instance:

If $\lambda$ is Lebesgue measure for $\mathbb {R}$ and π is Lebesgue measure for $\mathbb {R} ^{2}$ , then the product measure $\lambda \times \lambda =\pi .$ In terms of null sets, the following equivalence has been styled a Fubini's theorem:^[2]

Haar null[edit]

In a separable Banach space $(X,+),$ the group operation moves any subset $A\subseteq X$ to the translates $A+x$ for any $x\in X.$ When there is a probability measure $μ$ on the σ-algebra of Borel subsets of $X,$ such that for all $x,$ $\mu (A+x)=0,$ then $A$ is a Haar null set.^[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 $A^{-1}A$ contains an open neighborhood of the identity element.^[5] This property is named for Hugo Steinhaus since it is the conclusion of the Steinhaus theorem.