In measure theory, a null set is a set that is negligible for the purposes of the measure in question. Which sets are null will depend on the measure considered. Thus one may speak of m-null sets for a given measure m. In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a measure is a function that assigns a number, e. ...

The term "null set" is sometimes also used to refer to the empty set; see that article. Alternatively, it may be used for any notion of negligible set; see that article. Wikipedia uses the term "null set" only in the measure theoretic sense. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. ...

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 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 positives measures, this is equivalent to the definition given above; but for signed measures, this is stronger than simply saying that μ(N) = 0. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a measure is a function that assigns a number, e. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... 0 (zero), alternatively called naught or nought, is both a number and a numeral. ... In mathematics, the total variation of a real-valued function f on the bounded interval [a, b] is the supremum running over all partitions P = { x1, ..., xn } of the interval [a, b]. In effect, the total variation is the vertical component of the arc-length of the graph of f. ...

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. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

When talking about null sets in Euclidean n-space Rⁿ, it is usually understood that the measure being used is Lebesgue measure. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

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. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics the term countable set is used to describe the size of a set, e. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read sigma, means countable in this context) is a subset with certain desirable closure properties. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

In Lebesgue measure

For Lebesgue measure on Rⁿ, 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 Cantor set is an example of an uncountable null set in R. In mathematics, a singleton is a set with exactly one element. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the term dense has at least three different meanings. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, an uncountable set is a set which is not countable. ...

More generally, a subset N of R is null if and only if:

Given any positive number e, 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 e.

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. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... A negative number is a number that is less than zero, such as −3. ... there is is a song by boxcar racer ... This is a page about mathematics. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

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. The integral can be interpreted as the area under a curve. ...

A measure in which all null sets are measurable is complete. Any non-complete measure can be completed to form a complete measure by asserting that 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. In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0). ... In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b). ...