Null Set
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In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a null set N \subset \mathbb is a
measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
that has measure zero. This can be characterized as a set that can be
covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of ...
by a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
union of intervals of arbitrarily small total length. The notion of null set should not be confused with the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
as defined in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
. Although the empty set has
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
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 A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
M = (X, \Sigma, \mu) a null set is a set S\in\Sigma such that \mu(S) = 0.


Example

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 In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
is an example of an uncountable null set.


Definition

Suppose A is a subset of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
\mathbb such that \forall \varepsilon > 0, \ \exists \left\_n : U_n=(a_n,b_n)\subset \mathbb: \quad A \subset \bigcup_^\infty U_n \ \textrm\ \sum_^\infty \left, U_n\ < \varepsilon \,, where the are intervals and is the length of , then is a null set, also known as a set of zero-content. In terminology of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, this definition requires that there be a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of open covers of for which the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the lengths of the covers is zero.


Properties

The
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
is always a null set. More generally, any
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
union of null sets is null. Any 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 measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
.


Lebesgue measure

The
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
is the standard way of assigning a
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...
or
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
to subsets of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. A subset ''N'' of \mathbb has null Lebesgue measure and is considered to be a null set in \mathbb if and only if: : Given any positive number ''ε'', there is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of intervals in \mathbb such that ''N'' is contained in the union of the and the total length of the union is less than ''ε''. This condition can be generalised to \mathbb^, using ''n''-
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...
s instead of intervals. In fact, the idea can be made to make sense on any
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
, even if there is no Lebesgue measure there. For instance: * With respect to \mathbb^n, all
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, t ...
s are null, and therefore all
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
s are null. In particular, the set Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s is a null set, despite being dense in \mathbb. * The standard construction of the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
is an example of a null
uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
in \mathbb; however other constructions are possible which assign the Cantor set any measure whatsoever. * All the subsets of \mathbb^n whose
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
is smaller than ''n'' have null Lebesgue measure in \mathbb^n. For instance straight lines or circles are null sets in \mathbb^2. * Sard's lemma: the set of critical values of a smooth function has measure zero. If λ is Lebesgue measure for \mathbb and π is Lebesgue measure for \mathbb^, then the product measure \lambda \times \lambda = \pi. In terms of null sets, the following equivalence has been styled a Fubini's theorem: * For A \subset \mathbb^ and A_x = \ , \pi(A) = 0 \iff \lambda \left(\left\\right) = 0 .


Uses

Null sets play a key role in the definition of the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
: if functions and are equal except on a null set, then is integrable if and only if is, and their integrals are equal. This motivates the formal definition of spaces as sets of equivalence classes of functions which differ only on null sets. 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 is 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 In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
, which is closed hence Borel measurable, and which has measure zero, and to find a subset of which is not Borel measurable. (Since the Lebesgue measure is complete, this is of course Lebesgue measurable.) First, we have to know that every set of positive measure contains a nonmeasurable subset. Let be the Cantor function, a continuous function which is locally constant on , and monotonically increasing on , 1 with and . Obviously, is countable, since it contains one point per component of . Hence has measure zero, so has measure one. We need a strictly
monotonic function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
, so consider . Since is strictly monotonic and continuous, it is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
. Furthermore, has measure one. Let be non-measurable, and let . Because is injective, we have that , and so is a null set. However, if it were Borel measurable, then would also be Borel measurable (here we use the fact that the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or throug ...
of a Borel set by a continuous function is measurable; is the preimage of ''F'' through the continuous function .) Therefore, is a null, but non-Borel measurable set.


Haar null

In a separable
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
, the group operation moves any subset to the translates for any . When there is a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
on the σ-algebra of Borel subsets of , such that for all , , then is a Haar null set. 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 group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...
s have been related to the size of subsets and Haar null sets. Haar null sets have been used in Polish groups to show that when is not a
meagre set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
then contains an open neighborhood of the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
. This property is named for
Hugo Steinhaus Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Un ...
since it is the conclusion of the Steinhaus theorem.


See also

* Cantor function *
Measure (mathematics) In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
*
Empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
* Nothing


References


Further reading

* * * {{Measure theory Measure theory Set theory