Support (measure theory)

In mathematics, the support of a measure ${\displaystyle \mu }$ on a measure space ${\displaystyle (X,{\mathcal {A}},\mu )}$ is a precise notion of where in the space ${\displaystyle X}$ the measure "lives". It is defined to be the largest subset of ${\displaystyle X}$ on which the measure is strictly positive.

Motivation

Recall that the measure ${\displaystyle \mu }$  is really a function ${\displaystyle \mu :{\mathcal {A}}\to [0,+\infty ]}$ . Therefore, in terms of the usual definition of support, the support of ${\displaystyle \mu }$  is a subset of the sigma algebra ${\displaystyle {\mathcal {A}}}$ :

${\displaystyle \mathrm {supp} (\mu ):={\overline {\{A\in {\mathcal {A}}|\mu (A)>0\}}}.}$ 

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

1. Lebesgue measure ${\displaystyle \lambda }$  on the real line ${\displaystyle \mathbb {R} }$ . It seems clear that ${\displaystyle \lambda }$  "lives on" the whole of the real line.
2. A Dirac measure ${\displaystyle \delta _{p}}$  at some point ${\displaystyle p\in \mathbb {R} }$ . Again, intuition suggests that the measure ${\displaystyle \delta _{p}}$  "lives at" the point ${\displaystyle p}$ , 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:

1. We could remove the points where ${\displaystyle \mu }$  is zero, and take the support to be the remainder ${\displaystyle X\setminus \{x\in X|\mu (\{x\})=0\}}$ . This might work for the Dirac measure ${\displaystyle \delta _{p}}$ , but it would definitely not work for ${\displaystyle \lambda }$ : since the Lebesgue measure of any point is zero, this definition would give ${\displaystyle \lambda }$  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: ${\displaystyle \{x\in X|\exists \mathrm {open\,} N_{x}\ni x\mathrm {\,s.t\,} \mu (N_{x})>0\}}$  (or the closure of this). This is also too simplistic: by taking ${\displaystyle N_{x}=X}$  for all points ${\displaystyle x\in X}$ , this would make the support of every measure except the zero measure the whole of ${\displaystyle X}$ .

The idea of strict positivity is not too far from a workable definition:

Definition

Let ${\displaystyle (X,{\mathcal {T}})}$  be a topological space; let ${\displaystyle (X,{\mathcal {A}},\mu )}$  also be a measure space such that the sigma algebra ${\displaystyle {\mathcal {A}}}$  contains all open sets ${\displaystyle U\in {\mathcal {T}}}$ . Then the support of the measure ${\displaystyle \mu }$  is defined to be the set of all points ${\displaystyle x\in X}$  for which every open neighbourhood of ${\displaystyle x}$  has positive measure:

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

In other words, the support is the largest subset of ${\displaystyle X}$  (with respect to inclusion, ${\displaystyle \subseteq }$ ) on which the measure is strictly positive.

Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below.

Properties

• The support of a measure is closed in ${\displaystyle X}$ . Suppose that ${\displaystyle x}$  is a limit point of ${\displaystyle \mathrm {supp} (\mu )}$ , and let ${\displaystyle N_{x}}$  be an open neighbourhood of ${\displaystyle x}$ . Since ${\displaystyle x}$  is a limit point of the support, there is some ${\displaystyle y\in N_{x}\cap \mathrm {supp} (\mu )}$ , ${\displaystyle y\neq x}$ . But ${\displaystyle N_{x}}$  is also an open neighbourhood of ${\displaystyle y}$ , so ${\displaystyle \mu (N_{x})>0}$ , as required. Hence, ${\displaystyle \mathrm {supp} (\mu )}$  contains all its limit points, i.e. it is closed.
• If ${\displaystyle A}$  is a measurable set outside the support, then ${\displaystyle A}$  has measure zero:

${\displaystyle A\subseteq X\setminus \mathrm {supp} (\mu )\implies \mu (A)=0.}$ 

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

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

${\displaystyle \int _{X}f(x)\,\mathrm {d} \mu (x)=\int _{\mathrm {supp} (\mu )}f(x)\,\mathrm {d} \mu (x).}$ 

Examples

Lebesgue measure

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

Dirac measure

In the case of Dirac measure ${\displaystyle \delta _{p}}$ , let ${\displaystyle x\in \mathbb {R} }$  and consider two cases:

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

We conclude that ${\displaystyle \mathrm {supp} (\delta _{p})}$  is the closure of the singleton set ${\displaystyle \{p\}}$ , which is ${\displaystyle \{p\}}$  itself.

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

A uniform distribution

Consider the Borel measure ${\displaystyle \mu }$  on the real line defined by

${\displaystyle \mu (A):=\lambda (A\cap (0,1))}$ 

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