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In mathematics, the support of a measure <math>\mu</math> on a measurable topological space <math>(X, \mathrm{Borel}(X))</math> is a precise notion of where in the space <math>X</math> the measure "lives". It is defined to be the largest (closed) subset of <math>X</math> for which every open neighbourhood of every point of the set has positive measure.

Contents 1 Motivation 2 Definition 3 Properties 4 Examples 4.1 Lebesgue measure 4.2 Dirac measure 4.3 A uniform distribution 5 Signed and complex measures 6 Reference

[edit] Motivation

Recall that a (non-negative) measure <math>\mu</math> on a measurable space <math>(X, \mathcal{A})</math>is really a function <math>\mu : \mathcal{A} \to [0, + \infty]</math>. Therefore, in terms of the usual definition of support, the support of <math>\mu</math> is a subset of the sigma algebra <math>\mathcal{A}</math>:

<math>\mathrm{supp} (\mu) := \overline{\{ A \in \mathcal{A} | \mu (A) > 0 \}}.</math>

However, this definition is somewhat unsatisfactory: we do not even have a topology on <math>\mathcal{A}</math>! What we really want to know is where in the space <math>X</math> the measure <math>\mu</math> is non-zero. Consider two examples:

1. Lebesgue measure <math>\lambda</math> on the real line <math>\mathbb{R}</math>. It seems clear that <math>\lambda</math> "lives on" the whole of the real line.
2. A Dirac measure <math>\delta_{p}</math> at some point <math>p \in \mathbb{R}</math>. Again, intuition suggests that the measure <math>\delta_{p}</math> "lives at" the point <math>p</math>, 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:

1. We could remove the points where <math>\mu</math> is zero, and take the support to be the remainder <math>X \setminus \{ x \in X | \mu ( \{ x \} ) = 0 \}</math>. This might work for the Dirac measure <math>\delta_{p}</math>, but it would definitely not work for <math>\lambda</math>: since the Lebesgue measure of any point is zero, this definition would give <math>\lambda</math> 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:

<math>\{ x \in X | \mbox{for some open } N_{x} \ni x, \mu(N_{x}) > 0 \}</math>

(or the closure of this). This is also too simplistic: by taking <math>N_{x} = X</math> for all points <math>x \in X</math>, this would make the support of every measure except the zero measure the whole of <math>X</math>.

However, the idea of "local strict positivity" is not too far from a workable definition:

[edit] Definition

Let <math>(X, \mathcal{T})</math> be a topological space; let <math>\mathrm{Borel} (X)</math> denote the Borel σ-algebra on <math>X</math>,i.e. the smallest sigma algebra on <math>X</math> that contains all open sets <math>U \in \mathcal{T}</math>. Let <math>\mu</math> be a measure on <math>(X, \mathrm{Borel} (X))</math>. Then the support of <math>\mu</math> is defined to be the set of all points <math>x \in X</math> for which every open neighbourhood of <math>x</math> has positive measure:

<math>\mathrm{supp} (\mu) := \{ x \in X | \forall x \in N_{x} \in \mathcal{T}, \mu (N_{x}) > 0 \}.</math>

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 <math>C \subseteq X</math> (with respect to inclusion) such that

<math>U \in \mathcal{T} \mbox{ and } U \cap C \neq \emptyset \implies \mu (U \cap C) > 0,</math>

i.e. every open set that has non-trivial intersection with the support has positive measure.

[edit] Properties

• A measure <math>\mu</math> on <math>X</math> is strictly positive if and only if it has support <math>\mathrm{supp} (\mu) = X</math>. If <math>\mu</math> is strictly positive and <math>x \in X</math> is arbitrary, then any open neighbourhood of <math>x</math>, since it is an open set, has positive measure; hence, <math>x \in \mathrm{supp} (\mu)</math>, so <math>\mathrm{supp} (\mu) = X</math>. Conversely, if <math>\mathrm{supp} (\mu) = X</math>, 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, <math>\mu</math> is strictly positive.
• The support of a measure is closed in <math>X</math>. Suppose that <math>x</math> is a limit point of <math>\mathrm{supp} (\mu)</math>, and let <math>N_{x}</math> be an open neighbourhood of <math>x</math>. Since <math>x</math> is a limit point of the support, there is some <math>y \in N_{x} \cap \mathrm{supp} (\mu)</math>, <math>y \neq x</math>. But <math>N_{x}</math> is also an open neighbourhood of <math>y</math>, so <math>\mu (N_{x}) > 0</math>, as required. Hence, <math>\mathrm{supp} (\mu)</math> contains all its limit points, i.e. it is closed.
• If <math>A</math> is a measurable set outside the support, then <math>A</math> has measure zero:

<math>A \subseteq X \setminus \mathrm{supp} (\mu) \implies \mu (A) = 0.</math>

The converse is not true in general: it fails if there exists <math>x \in \mathrm{supp} (\mu)</math> such that <math>\mu \left( \{ x \} \right) = 0</math> (e.g. Lebesgue measure).

• One does not need to "integrate outside the support": for any measurable function <math>f : X \to \mathbb{R}</math> or <math>\mathbb{C}</math>,

<math>\int_{X} f(x) \, \mathrm{d} \mu (x) = \int_{\mathrm{supp} (\mu)} f(x) \, \mathrm{d} \mu (x).</math>

[edit] Examples

[edit] Lebesgue measure

In the case of Lebesgue measure on the real line, consider an arbitrary point <math>x \in \mathbb{R}</math>. Then any open neighbourhood <math>N_{x}</math> of <math>x</math> must contain some open interval <math>(x - \varepsilon, x + \varepsilon)</math> for some <math>\varepsilon > 0</math>. This interval has Lebesgue measure <math>\varepsilon > 0</math>, so <math>\mu (N_{x}) \geq \varepsilon > 0</math>. Since <math>x \in \mathbb{R}</math> was arbitrary, <math>\mathrm{supp} (\lambda) = \mathbb{R}</math>.

[edit] Dirac measure

In the case of Dirac measure <math>\delta_{p}</math>, let <math>x \in \mathbb{R}</math> and consider two cases:

1. if <math>x = p</math>, then every open neighbourhood <math>N_{x}</math> of <math>x</math> contains <math>p</math>, so <math>\delta_{p} (N_{x}) = 1 > 0</math>;
2. on the other hand, if <math>x \neq p</math>, then there exists a sufficiently small open ball <math>B</math> around <math>x</math> that does not contain <math>p</math>, so <math>\delta_{p} (B) = 0</math>.

We conclude that <math>\mathrm{supp} (\delta_{p})</math> is the closure of the singleton set <math>\{ p \}</math>, which is <math>\{ p \}</math> itself.

In fact, a measure <math>\mu</math> on the real line is a Dirac measure <math>\delta_{p}</math> for some point <math>p</math> if and only if the support of <math>\mu</math> is the singleton set <math>\{ p \}</math>. Consequently, Dirac measure on the real line is the unique measure with zero variance [provided that the measure has variance at all].

[edit] A uniform distribution

Consider the measure <math>\mu</math> on the real line defined by

<math>\mu (A) := \lambda (A \cap (0, 1))</math>

i.e. a uniform measure on the open interval <math>(0, 1)</math>. A similar argument to the Dirac measure example shows that <math>\mathrm{supp} (\mu) = [0, 1]</math>. 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 <math>(0, 1)</math>, and so must have positive <math>\mu</math>-measure.

[edit] Signed and complex measures

Suppose that <math>\mu : \mathcal{A} \to [- \infty, + \infty]</math> is a signed measure. Use the Hahn decomposition theorem to write

<math>\mu = \mu^{+} - \mu^{-},</math>

where <math>\mu^{\pm}</math> are both non-negative measures. Then the support of <math>\mu</math> is defined to be

<math>\mathrm{supp} (\mu) := \mathrm{supp} (\mu^{+}) \cup \mathrm{supp} (\mu^{-}).</math>

Similarly, if <math>\mu : \mathcal{A} \to \mathbb{C}</math> is a complex measure, the support of <math>\mu</math> is defined to be the union of the supports of its real and imaginary parts.

[edit] 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 3-764-32428-7.pl:Nośnik miary