
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 ( ...
, 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
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 (mathematics), set is countable if either it is finite set, 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 fro ...
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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
as defined in
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
. 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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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 ...
a null set is a set
such that
Examples
Every finite or
countably infinite
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 numbe ...
subset of the
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
is a null set. For example, the set of
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
, the set of
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
and the set of
algebraic numbers are all 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. It is uncountable because it contains all real numbers between 0 and 1 whose
ternary expansion can be written using only 0’s and 2’s (see
Cantor's diagonal argument
Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infin ...
), and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and iteratively removing a third of the previous set, thereby multiplying the length by 2/3 with every step.
Definition for 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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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 (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
,
area
Area is the measure of a region's size on a 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 surface or the boundary of a three-di ...
or
volume
Volume is a measure of regions in 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) ...
to subsets of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
.
A subset
of the
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
has null Lebesgue measure and is considered to be a null set (also known as a set of zero-content) in
if and only if:
:
Given any
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by e ...
positive number
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
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 cal ...
of
intervals in
(where interval
has length
) such that
is contained in the union of the
and the total length of the union is less than
i.e.,
(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 ( ...
, 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 cal ...
of
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...
s of
for which the
limit of the lengths of the covers is zero.)
This condition can be generalised to
using
-
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
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 N ...
, even if there is no Lebesgue measure there.
For instance:
* With respect to
all
singleton sets 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 numbe ...
s are null. In particular, the set
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 (for example,
The set of all ...
s is a null set, despite being
dense in
* The standard construction of the
Cantor set is an example of a null
uncountable set
In mathematics, an uncountable set, informally, 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 number is larger t ...
in
however other constructions are possible which assign the Cantor set any measure whatsoever.
* All the subsets of
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 coo ...
is smaller than
have null Lebesgue measure in
For instance straight lines or circles are null sets in
*
Sard's lemma: the set of critical values of a smooth function has measure zero.
If
is Lebesgue measure for
and π is Lebesgue measure for
, then the
product measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology o ...
In terms of null sets, the following equivalence has been styled a
Fubini's theorem:
* For
and
Measure-theoretic properties
Let
be a
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 ...
. We have:
*
(by
definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
of
).
* Any
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
union of null sets is itself a null set (by
countable subadditivity of
).
* Any (measurable) subset of a null set is itself a null set (by
monotonicity of
).
Together, these facts show that the null sets of
form a
𝜎-ideal of the
𝜎-algebra . Accordingly, null sets may be interpreted as
negligible sets, yielding a measure-theoretic 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 to ...
".
Uses
Null sets play a key role in the definition of the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
: 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
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
.
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 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
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 or ...
, so consider
Since
is strictly monotonic and continuous, it is a
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
. 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 of a Borel set by a continuous function is measurable;
is the preimage of
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 (, ) 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 vectors and ...
addition 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 σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
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
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfr� ...
.
Some algebraic properties of
topological group
In mathematics, topological groups are 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 structures ...
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 then
contains an open neighborhood of the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
.
This property is named for
Hugo Steinhaus since it is the conclusion of the
Steinhaus theorem.
References
Further reading
*
*
*
{{Measure theory
Measure theory
Set theory