In mathematics, the support 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a measure is a function that assigns a number, e. ... 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. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Motivation

Recall that a (non-negative) measure μ on a measurable space (X, Σ) is really a function μ : Σ → [0, +∞]. Therefore, in terms of the usual definition of support, the support of μ is a subset of the σ-algebra Σ: A definition is a concise statement explaining the meaning of a term, word or phrase. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...

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:

1. Lebesgue measure λ on the real line R. It seems clear that λ "lives on" the whole of the real line.
2. A Dirac measure δ_p at some point p ∈ R. Again, intuition suggests that the measure δ_p "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: In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that δx({x}) = 1 and in general δx(Y) = 0 for any subset Y of X not containing x, δx(Z) = 1 for any...

1. 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 δ_p, but it would definitely not work for λ: since the Lebesgue measure of any point is zero, this definition would give λ empty support.
2. By comparison with the notion of strict positivity of measures, we could take the support to be the set of all points with a neighbourhood of positive measure:

(or the closure of this). This is also too simplistic: by taking N_x = 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: In mathematics, strict positivity is a concept in measure theory. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

Definition

Let (X, T) be a topological space; 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 of μ is defined to be the set of all points x in X for which every open neighbourhood of x has positive measure: Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ...

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 In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

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 if 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 set, has positive measure; hence, x ∈ supp(μ), so supp(μ) = X. Conversely, if supp(μ) = X, then every open set is an open neighbourhhod of some point in its interior, which is also a point of the support, and so has positive measure; hence, μ is strictly positive.
• The support of a measure is closed in X. Suppose that x is a limit point of supp(μ), and let N_x be an open neighbourhood of x. Since x is a limit point of the support, there is some y ∈ N_x ∩ supp(μ), y ≠ x. But N_x is also an open neighbourhood of y, so μ(N_x) > 0, as required. Hence, supp(μ) contains all its limit points, i.e. it is closed.
• If A is a measurable set outside the support, then A 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).

• One does not need to "integrate outside the support": for any measurable function f : X → R or C,

It has been suggested that this article or section be merged with Logical biconditional. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... In mathematics, a measure is a function that assigns a number, e. ... Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

Examples

Lebesgue measure

In the case of Lebesgue measure λ on the real line R, consider an arbitrary point x ∈ R. Then any open neighbourhood N_x of x must contain some open interval (x − ε, x + ε) for some ε > 0. This interval has Lebesgue measure 2ε > 0, so λ(N_x) ≥ 2ε > 0. Since x ∈ R was arbitrary, supp(λ) = R. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Dirac measure

In the case of Dirac measure δ_p, let x ∈ R and consider two cases:

1. if x = p, then every open neighbourhood N_x of x contains p, so δ_p(N_x) = 1 > 0;
2. on the other hand, if x ≠ p, then there exists a sufficiently small open ball B around x that does not contain p, so δ_p(B) = 0.

We conclude that supp(δ_p) is the closure of the singleton set {p}, which is {p} itself. In mathematics, a singleton is a set with exactly one element. ...

In fact, a measure μ on the real line is a Dirac measure δ_p 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]. It has been suggested that this article or section be merged with Logical biconditional. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

A uniform distribution

Consider the measure μ on the real line R defined by

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. In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ...

Signed and complex measures

Suppose that μ : Σ → [−∞, +∞] is a signed measure. Use the Hahn decomposition theorem to write In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. ... The Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space (X,Σ) and a signed measure μ defined on the σ-algebra Σ, there exist two sets P and N in Σ such that: P∪N = X and P∩N = ∅. For each E in Σ such that E ⊆ P...

μ = μ ⁺ − μ ⁻ ,

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. In mathematics, or more specifically in measure theory, a complex measure is a generalisation of the concept of measure by letting it have complex values. ... 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. ...

Reference

• Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 3-7643-2428-7.