Support (measure theory)
Encyclopedia
In mathematics
, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space
(X, Borel(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest (closed
) subset
of X for which every open
neighbourhood
of every point of the set has positive measure.
of support
, the support of μ is a subset of the σ-algebra Σ:
However, this definition is somewhat unsatisfactory: we do not even have a topology on Σ! What we really want to know is where in the space X the measure μ is non-zero. Consider two examples:
In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section:
|closure]] of this). It is also too simplistic: by taking Nx = X for all points x ∈ X, this would make the support of every measure except the zero measure the whole of X.
However, the idea of "local strict positivity" is not too far from a workable definition:
; let Borel(X) denote the Borel σ-algebra
on X, i.e. the smallest sigma algebra on X that contains all open sets U ∈ T. Let μ be a measure on (X, Borel(X)). Then the support (or spectrum) of μ is defined to be the set of all points x in X for which every open
neighbourhood
Nx of x has positive
measure:
Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below. As such, an equivalent definition of the support is as the largest closed set
C ⊆ X (with respect to inclusion) such that
i.e. every open set that has non-trivial intersection with the support has positive measure.
(x − ε, x + ε) for some ε > 0. This interval has Lebesgue measure 2ε > 0, so λ(Nx) ≥ 2ε > 0. Since x ∈ R was arbitrary, supp(λ) = R.
We conclude that supp(δp) is the closure of the singleton set {p}, which is {p} itself.
In fact, a measure μ on the real line is a Dirac measure δp for some point p if and only if
the support of μ is the singleton set {p}. Consequently, Dirac measure on the real line is the unique measure with zero variance
[provided that the measure has variance at all].
i.e. a uniform measure
on the open interval (0, 1). A similar argument to the Dirac measure example shows that supp(μ) = [0, 1]. Note that the boundary points 0 and 1 lie in the support: any open set containing 0 (or 1) contains an open interval about 0 (or 1), which must intersect (0, 1), and so must have positive μ-measure.
. Use the Hahn decomposition theorem
to write
where μ± are both non-negative measures. Then the support of μ is defined to be
Similarly, if μ : Σ → C is a complex measure
, the support of μ is defined to be the union
of the supports of its real and imaginary parts.
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...
, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(X, Borel(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest (closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
) 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 X for which every open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
neighbourhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of every point of the set has positive measure.
Motivation
A (non-negative) measure μ on a measurable space (X, Σ) is really a function μ : Σ → [0, +∞]. Therefore, in terms of the usual definitionDefinition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...
of support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
, the support of μ is a subset of the σ-algebra Σ:
However, this definition is somewhat unsatisfactory: we do not even have a topology on Σ! What we really want to know is where in the space X the measure μ is non-zero. Consider two examples:
- Lebesgue measureLebesgue measureIn 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...
λ on the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R. It seems clear that λ "lives on" the whole of the real line. - A Dirac measure δp at some point p ∈ R. Again, intuition suggests that the measure δp "lives at" the point p, and nowhere else.
In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section:
- We could remove the points where μ is zero, and take the support to be the remainder X \ { x ∈ X | μ({x}) = 0 }. This might work for the Dirac measure δp, but it would definitely not work for λ: since the Lebesgue measure of any point is zero, this definition would give λ empty support.
- By comparison with the notion of strict positivityStrictly positive measureIn mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that it is zero "only on points".-Definition:...
of measures, we could take the support to be the set of all points with a neighbourhood of positive measure:
|closure]] of this). It is also too simplistic: by taking Nx = X for all points x ∈ X, this would make the support of every measure except the zero measure the whole of X.
However, the idea of "local strict positivity" is not too far from a workable definition:
Definition
Let (X, T) be a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
; let Borel(X) denote the Borel σ-algebra
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
on X, i.e. the smallest sigma algebra on X that contains all open sets U ∈ T. Let μ be a measure on (X, Borel(X)). Then the support (or spectrum) of μ is defined to be the set of all points x in X for which every open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
neighbourhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
Nx of x has positive
Positive
Positive is a property of positivity and may refer to:- Mathematics and science :* Converging lens or positive lens, in optics* Plus sign, the sign "+" used to indicate a positive number* Positive , a polarity of electrical charge...
measure:
Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below. As such, an equivalent definition of the support is as the largest closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
C ⊆ X (with respect to inclusion) such that
i.e. every open set that has non-trivial intersection with the support has positive measure.
Properties
- A measure μ on X is strictly positive if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it has support supp(μ) = X. If μ is strictly positive and x ∈ X is arbitrary, then any open neighbourhood of x, since it is an open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
, has positive measure; hence, x ∈ supp(μ), so supp(μ) = X. Conversely, if supp(μ) = X, then every open set (being an open neighbourhood of some point in its interior, which is also a point of the support) has positive measure; hence, μ is strictly positive. - The support of a measure is closedClosed setIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
in X. Suppose that x is a limit pointLimit pointIn mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
of supp(μ), and let Nx be an open neighbourhood of x. Since x is a limit point of the support, there is some y ∈ Nx ∩ supp(μ), y ≠ x. But Nx is also an open neighbourhood of y, so μ(Nx) > 0, as required. Hence, supp(μ) contains all its limit points, i.e. it is closed. Alternatively, let y be a point outside supp(μ). By definition, there is a neighbourhood Ny of y with measure 0. No point of this neighbourhood belongs to supp(μ), thus Ny does not intersect the support. This means the complement of supp(μ) is locally open, therefore open. - Under certain conditions on X and µ, for instance X being a topological Hausdorff spaceHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and µ being a Radon measureRadon measureIn mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
, a measurable set A outside the support has measure zero:
-
- The converse is not true in general: it fails if there exists a point x ∈ supp(μ) such that μ({x}) = 0 (e.g. Lebesgue measure).
- Thus, one does not need to "integrate outside the support": for any measurable functionMeasurable functionIn mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
f : X → R or C,
Lebesgue measure
In the case of Lebesgue measure λ on the real line R, consider an arbitrary point x ∈ R. Then any open neighbourhood Nx of x must contain some open intervalInterval (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...
(x − ε, x + ε) for some ε > 0. This interval has Lebesgue measure 2ε > 0, so λ(Nx) ≥ 2ε > 0. Since x ∈ R was arbitrary, supp(λ) = R.
Dirac measure
In the case of Dirac measure δp, let x ∈ R and consider two cases:- if x = p, then every open neighbourhood Nx of x contains p, so δp(Nx) = 1 > 0;
- on the other hand, if x ≠ p, then there exists a sufficiently small open ball B around x that does not contain p, so δp(B) = 0.
We conclude that supp(δp) is the closure of the singleton set {p}, which is {p} itself.
In fact, a measure μ on the real line is a Dirac measure δp for some point p if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the support of μ is the singleton set {p}. Consequently, Dirac measure on the real line is the unique measure with zero variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
[provided that the measure has variance at all].
A uniform distribution
Consider the measure μ on the real line R defined byi.e. a uniform measure
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
on the open interval (0, 1). A similar argument to the Dirac measure example shows that supp(μ) = [0, 1]. Note that the boundary points 0 and 1 lie in the support: any open set containing 0 (or 1) contains an open interval about 0 (or 1), which must intersect (0, 1), and so must have positive μ-measure.
Signed and complex measures
Suppose that μ : Σ → [−∞, +∞] is a signed measureSigned measure
In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...
. Use the Hahn decomposition theorem
Hahn decomposition theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space and a signed measure μ defined on the σ-algebra Σ, there exist two sets P and N in Σ such that:...
to write
where μ± are both non-negative measures. Then the support of μ is defined to be
Similarly, if μ : Σ → C is a complex measure
Complex measure
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size is a complex number.-Definition:...
, the support of μ is defined to be the 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 the supports of its real and imaginary parts.