# Null set

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In mathematics, a **null set** is a set that is negligible *in some sense*. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set (or simply a **measure-zero set**). More generally, whenever an ideal is taken as understood, then a null set is any element of that ideal.

The remainder of this article discusses the measure-theoretic notion.

## Contents

## Definition

Let *X* be a measurable space, let μ be a measure on *X*, and let *N* be a measurable set in *X*. If μ is a positive measure, then *N* is null (or *zero measure*) if its measure μ(*N*) is zero. If μ is not a positive measure, then *N* is μ-null if *N* is |μ|-null, where |μ| is the total variation of μ; equivalently, if every measurable subset *A* of *N* satisfies μ(*A*) = 0. For positive measures, this is equivalent to the definition given above; but for signed measures, this is stronger than simply saying that μ(*N*) = 0.

A nonmeasurable set is considered null if it is 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 **R**^{n}, 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 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.

## Lebesgue measure

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

A subset *N* of **R** has null Lebesgue measure and is considered to be a null set in **R** if and only if:

- Given any positive number
*ε*, there is a sequence {*I*_{n}} of intervals such that*N*is contained in the union of the*I*_{n}and the total length of the*I*_{n}is less than*ε*.

This condition can be generalised to **R**^{n}, 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.

For instance:

- With respect to
**R**^{n}, all 1-point sets 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 standard construction of the Cantor set is an example of a null uncountable set in
**R**; however other constructions are possible which assign the Cantor set any measure whatsoever. - All the subsets of
**R**^{n}whose dimension is smaller than*n*have null Lebesgue measure in**R**^{n}. For instance straight lines or circles are null sets in**R**^{2}. - Sard's lemma: the set of critical values of a smooth function has measure zero.

## 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 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's defined as the completion of a non-complete Borel measure.

### A subset of the Cantor set which is not Borel measurable

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 *K ^{c}*, and monotonically increasing on [0, 1], with

*f*(0) = 0 and

*f*(1) = 1. Obviously,

*f*(

*K*) is countable, since it contains one point per component of

^{c}*K*. Hence

^{c}*f*(

*K*) has measure zero, so

^{c}*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.