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.

In some elementary textbooks, null set is taken to mean empty set.

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 Rn, 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:

This condition can be generalised to Rn, 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.

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