Null set
Encyclopedia
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.
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
.
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 set
s, defining a notion of almost everywhere
.
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.
For instance:
: 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.
, 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 the unit interval, 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
, so consider
. Since 
is strictly monotonic and continuous, it is a homeomorphism
. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Ideal (set theory)
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal , and the union of any two elements of the ideal must also be in the ideal.More formally, given a set X, an...
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
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
.
The remainder of this article discusses the measure-theoretic notion.
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
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
.
If μ is not a positive measure, then N is μ-null if N is |μ|-null, where |μ| is the total variation
Total variation
In mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure...
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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn, it is usually understood that the measure being used is Lebesgue measure
Lebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
.
Properties
The empty setEmpty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
is always a null set.
More generally, any countable union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
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
Sigma-ideal
In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.Let be a measurable space...
on X.
Similarly, the measurable m-null sets form a sigma-ideal of the sigma-algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
of measurable sets.
Thus, null sets may be interpreted as negligible set
Negligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.Negligible sets define...
s, defining a notion of almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
.
Lebesgue measure
The Lebesgue measureLebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
is the standard way of assigning a length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
, area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
or volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
to subsets of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
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 isExistential quantificationIn predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...
a sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
{In} of intervalsInterval (mathematics)In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
such that N is contained in the union of the In and the total length of the In is less than ε.
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
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
, even if there is no Lebesgue measure there.
For instance:
- With respect to Rn, all 1-point sets are null, and therefore all countable setCountable setIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
s are null. In particular, the set Q of rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s is a null set, despite being dense in R. - The Cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
is an example of a null uncountable setUncountable setIn mathematics, an uncountable 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 number is larger than that of the set of all natural numbers.-Characterizations:There...
in R. - All the subsets of Rn whose dimensionDimensionIn physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
is smaller than n have null Lebesgue measure in Rn. For instance straight lines or circles are null sets in R2. - Sard's lemmaSard's lemmaSard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set...
: the set of critical valueCritical value-Differential topology:In differential topology, a critical value of a differentiable function between differentiable manifolds is the image ƒ in N of a critical point x in M.The basic result on critical values is Sard's lemma...
s of a smooth function has measure zero.
Uses
Null sets play a key role in the definition of the Lebesgue integralLebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
: 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
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
.
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 setCantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
, 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 functionCantor function
In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:See figure...
, a continuous function which is locally constant on
, and monotonically increasing on the unit interval, 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 functionMonotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
, so consider
. Since is strictly monotonic and continuous, it is a homeomorphismHomeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
. 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.