A (non-negative) measure ${\displaystyle \mu }$  on a measurable space ${\displaystyle (X,\Sigma )}$  is really a function ${\displaystyle \mu :\Sigma \to [0,+\infty ].}$  Therefore, in terms of the usual definition of support, the support of ${\displaystyle \mu }$  is a subset of the σ-algebra ${\displaystyle \Sigma :}$ 

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 ${\displaystyle X\setminus \{x\in X\mid \mu (\{x\})=0\}.}$  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 singleton 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:
   $${\displaystyle \{x\in X\mid \exists N_{x}{\text{ open}}{\text{ such that }}(x\in N_{x}{\text{ and }}\mu (N_{x})>0)\}}$$

    
   (or the closure of this). It 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 ${\displaystyle X.}$ 

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

Let ${\displaystyle (X,T)}$  be a topological space; let ${\displaystyle B(T)}$  denote the Borel σ-algebra on ${\displaystyle X,}$  i.e. the smallest sigma algebra on ${\displaystyle X}$  that contains all open sets ${\displaystyle U\in T.}$  Let ${\displaystyle \mu }$  be a measure on ${\displaystyle (X,B(T))}$  Then the support (or spectrum) of ${\displaystyle \mu }$  is defined as the set of all points ${\displaystyle x}$  in ${\displaystyle X}$  for which every open neighbourhood ${\displaystyle N_{x}}$  of ${\displaystyle x}$  has positive measure:

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

An equivalent definition of support is as the largest ${\displaystyle C\in B(T)}$  (with respect to inclusion) such that every open set which has non-empty intersection with ${\displaystyle C}$  has positive measure, i.e. the largest ${\displaystyle C}$  such that:

Signed and complex measures edit

This definition can be extended to signed and complex measures. Suppose that ${\displaystyle \mu :\Sigma \to [-\infty ,+\infty ]}$  is a signed measure. Use the Hahn decomposition theorem to write

Similarly, if ${\displaystyle \mu :\Sigma \to \mathbb {C} }$  is a complex measure, the support of ${\displaystyle \mu }$  is defined to be the union of the supports of its real and imaginary parts.

${\displaystyle \operatorname {supp} (\mu _{1}+\mu _{2})=\operatorname {supp} (\mu _{1})\cup \operatorname {supp} (\mu _{2})}$  holds.

A measure ${\displaystyle \mu }$  on ${\displaystyle X}$  is strictly positive if and only if it has support ${\displaystyle \operatorname {supp} (\mu )=X.}$  If ${\displaystyle \mu }$  is strictly positive and ${\displaystyle x\in X}$  is arbitrary, then any open neighbourhood of ${\displaystyle x,}$  since it is an open set, has positive measure; hence, ${\displaystyle x\in \operatorname {supp} (\mu ),}$  so ${\displaystyle \operatorname {supp} (\mu )=X.}$  Conversely, if ${\displaystyle \operatorname {supp} (\mu )=X,}$  then every non-empty open set (being an open neighbourhood of some point in its interior, which is also a point of the support) has positive measure; hence, ${\displaystyle \mu }$  is strictly positive. The support of a measure is closed in ${\displaystyle X,}$ as its complement is the union of the open sets of measure ${\displaystyle 0.}$ 

In general the support of a nonzero measure may be empty: see the examples below. However, if ${\displaystyle X}$  is a Hausdorff topological space and ${\displaystyle \mu }$  is a Radon measure, a Borel set ${\displaystyle A}$  outside the support has measure zero:

The concept of support of a measure and that of spectrum of a self-adjoint linear operator on a Hilbert space are closely related. Indeed, if ${\displaystyle \mu }$  is a regular Borel measure on the line ${\displaystyle \mathbb {R} ,}$  then the multiplication operator ${\displaystyle (Af)(x)=xf(x)}$  is self-adjoint on its natural domain

Lebesgue measure edit

In the case of Lebesgue measure ${\displaystyle \lambda }$  on the real line ${\displaystyle \mathbb {R} ,}$  consider an arbitrary point ${\displaystyle x\in \mathbb {R} .}$  Then any open neighbourhood ${\displaystyle N_{x}}$  of ${\displaystyle x}$  must contain some open interval ${\displaystyle (x-\epsilon ,x+\epsilon )}$  for some ${\displaystyle \epsilon >0.}$  This interval has Lebesgue measure ${\displaystyle 2\epsilon >0,}$  so ${\displaystyle \lambda (N_{x})\geq 2\epsilon >0.}$  Since ${\displaystyle x\in \mathbb {R} }$  was arbitrary, ${\displaystyle \operatorname {supp} (\lambda )=\mathbb {R} .}$ 

Dirac measure edit

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

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 ${\displaystyle \operatorname {supp} (\delta _{p})}$  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 edit

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

A nontrivial measure whose support is empty edit

The space of all countable ordinals with the topology generated by "open intervals" is a locally compact Hausdorff space. The measure ("Dieudonné measure") that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.

A nontrivial measure whose support has measure zero edit

On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure ${\displaystyle 0.}$  An example of this is given by adding the first uncountable ordinal ${\displaystyle \Omega }$  to the previous example: the support of the measure is the single point ${\displaystyle \Omega ,}$  which has measure ${\displaystyle 0.}$ 

• 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276. ISBN 0-8218-3889-X. MR2169627 (See chapter 2, section 2.)
• Teschl, Gerald (2009). Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS.(See chapter 3, section 2)