Support (measure theory)

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

A (non-negative) measure ${\displaystyle \mu }${\displaystyle \mu } on a measurable space ${\displaystyle (X,\Sigma )}${\displaystyle (X,\Sigma )} is really a function ${\displaystyle \mu :\Sigma \to [0,+\infty ].}${\displaystyle \mu :\Sigma \to [0,+\infty ].} Therefore, in terms of the usual definition of support, the support of ${\displaystyle \mu }${\displaystyle \mu } is a subset of the σ-algebra ${\displaystyle \Sigma :}${\displaystyle \Sigma :} $${\displaystyle \operatorname {supp} (\mu ):={\overline {\{A\in \Sigma ,円\vert ,円\mu (A)\neq 0\}}},}$${\displaystyle \operatorname {supp} (\mu ):={\overline {\{A\in \Sigma ,円\vert ,円\mu (A)\neq 0\}}},} where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on ${\displaystyle \Sigma .}${\displaystyle \Sigma .} What we really want to know is where in the space ${\displaystyle X}${\displaystyle X} the measure ${\displaystyle \mu }${\displaystyle \mu } is non-zero. Consider two examples:

1. Lebesgue measure ${\displaystyle \lambda }${\displaystyle \lambda } on the real line ${\displaystyle \mathbb {R} .}${\displaystyle \mathbb {R} .} It seems clear that ${\displaystyle \lambda }${\displaystyle \lambda } "lives on" the whole of the real line.
2. A Dirac measure ${\displaystyle \delta _{p}}${\displaystyle \delta _{p}} at some point ${\displaystyle p\in \mathbb {R} .}${\displaystyle p\in \mathbb {R} .} Again, intuition suggests that the measure ${\displaystyle \delta _{p}}${\displaystyle \delta _{p}} "lives at" the point ${\displaystyle p,}${\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 ${\displaystyle \mu }${\displaystyle \mu } is zero, and take the support to be the remainder ${\displaystyle X\setminus \{x\in X\mid \mu (\{x\})=0\}.}${\displaystyle X\setminus \{x\in X\mid \mu (\{x\})=0\}.} This might work for the Dirac measure ${\displaystyle \delta _{p},}${\displaystyle \delta _{p},} but it would definitely not work for ${\displaystyle \lambda :}${\displaystyle \lambda :} since the Lebesgue measure of any singleton is zero, this definition would give ${\displaystyle \lambda }${\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: $${\displaystyle \{x\in X\mid \exists N_{x}{\text{ open}}{\text{ such that }}(x\in N_{x}{\text{ and }}\mu (N_{x})>0)\}}$${\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 ${\displaystyle N_{x}=X}${\displaystyle N_{x}=X} for all points ${\displaystyle x\in X,}${\displaystyle x\in X,} this would make the support of every measure except the zero measure the whole of ${\displaystyle X.}${\displaystyle X.}

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

Let ${\displaystyle (X,T)}${\displaystyle (X,T)} be a topological space; let ${\displaystyle B(T)}${\displaystyle B(T)} denote the Borel σ-algebra on ${\displaystyle X,}${\displaystyle X,} i.e. the smallest sigma algebra on ${\displaystyle X}${\displaystyle X} that contains all open sets ${\displaystyle U\in T.}${\displaystyle U\in T.} Let ${\displaystyle \mu }${\displaystyle \mu } be a measure on ${\displaystyle (X,B(T))}${\displaystyle (X,B(T))}. Then the support (or spectrum) of ${\displaystyle \mu }${\displaystyle \mu } is defined as the set of all points ${\displaystyle x}${\displaystyle x} in ${\displaystyle X}${\displaystyle X} for which every open neighbourhood ${\displaystyle N_{x}}${\displaystyle N_{x}} of ${\displaystyle x}${\displaystyle x} has positive measure: $${\displaystyle \operatorname {supp} (\mu ):=\{x\in X\mid \forall N_{x}\in T\colon (x\in N_{x}\Rightarrow \mu (N_{x})>0)\}.}$${\displaystyle \operatorname {supp} (\mu ):=\{x\in X\mid \forall N_{x}\in T\colon (x\in N_{x}\Rightarrow \mu (N_{x})>0)\}.}

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 ${\displaystyle C\in B(T)}${\displaystyle C\in B(T)} (with respect to inclusion) such that every open set which has non-empty intersection with ${\displaystyle C}${\displaystyle C} has positive measure, i.e. the largest ${\displaystyle C}${\displaystyle C} such that: $${\displaystyle (\forall U\in T)(U\cap C\neq \varnothing \implies \mu (U\cap C)>0).}$${\displaystyle (\forall U\in T)(U\cap C\neq \varnothing \implies \mu (U\cap C)>0).}

This definition can be extended to signed and complex measures. Suppose that ${\displaystyle \mu :\Sigma \to [-\infty ,+\infty ]}${\displaystyle \mu :\Sigma \to [-\infty ,+\infty ]} is a signed measure. Use the Hahn decomposition theorem to write $${\displaystyle \mu =\mu ^{+}-\mu ^{-},}$${\displaystyle \mu =\mu ^{+}-\mu ^{-},} where ${\displaystyle \mu ^{\pm }}${\displaystyle \mu ^{\pm }} are both non-negative measures. Then the support of ${\displaystyle \mu }${\displaystyle \mu } is defined to be $${\displaystyle \operatorname {supp} (\mu ):=\operatorname {supp} (\mu ^{+})\cup \operatorname {supp} (\mu ^{-}).}$${\displaystyle \operatorname {supp} (\mu ):=\operatorname {supp} (\mu ^{+})\cup \operatorname {supp} (\mu ^{-}).}

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

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

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

In general the support of a nonzero measure may be empty: see the examples below. However, if ${\displaystyle X}${\displaystyle X} is a Hausdorff topological space and ${\displaystyle \mu }${\displaystyle \mu } is a Radon measure, a Borel set ${\displaystyle A}${\displaystyle A} outside the support has measure zero: $${\displaystyle A\subseteq X\setminus \operatorname {supp} (\mu )\implies \mu (A)=0.}$${\displaystyle A\subseteq X\setminus \operatorname {supp} (\mu )\implies \mu (A)=0.} The converse is true if ${\displaystyle A}${\displaystyle A} is open, but it is not true in general: it fails if there exists a point ${\displaystyle x\in \operatorname {supp} (\mu )}${\displaystyle x\in \operatorname {supp} (\mu )} such that ${\displaystyle \mu (\{x\})=0}${\displaystyle \mu (\{x\})=0} (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any measurable function ${\displaystyle f:X\to \mathbb {R} }${\displaystyle f:X\to \mathbb {R} } or ${\displaystyle \mathbb {C} ,}${\displaystyle \mathbb {C} ,} $${\displaystyle \int _{X}f(x),円\mathrm {d} \mu (x)=\int _{\operatorname {supp} (\mu )}f(x),円\mathrm {d} \mu (x).}$${\displaystyle \int _{X}f(x),円\mathrm {d} \mu (x)=\int _{\operatorname {supp} (\mu )}f(x),円\mathrm {d} \mu (x).}

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 ${\displaystyle \mu }${\displaystyle \mu } is a regular Borel measure on the line ${\displaystyle \mathbb {R} ,}${\displaystyle \mathbb {R} ,} then the multiplication operator ${\displaystyle (Af)(x)=xf(x)}${\displaystyle (Af)(x)=xf(x)} is self-adjoint on its natural domain $${\displaystyle D(A)=\{f\in L^{2}(\mathbb {R} ,d\mu )\mid xf(x)\in L^{2}(\mathbb {R} ,d\mu )\}}$${\displaystyle D(A)=\{f\in L^{2}(\mathbb {R} ,d\mu )\mid xf(x)\in L^{2}(\mathbb {R} ,d\mu )\}} and its spectrum coincides with the essential range of the identity function ${\displaystyle x\mapsto x,}${\displaystyle x\mapsto x,} which is precisely the support of ${\displaystyle \mu .}${\displaystyle \mu .}^[1]

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

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

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

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

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

Consider the measure ${\displaystyle \mu }${\displaystyle \mu } on the real line ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } defined by $${\displaystyle \mu (A):=\lambda (A\cap (0,1))}$${\displaystyle \mu (A):=\lambda (A\cap (0,1))} i.e. a uniform measure on the open interval ${\displaystyle (0,1).}${\displaystyle (0,1).} A similar argument to the Dirac measure example shows that ${\displaystyle \operatorname {supp} (\mu )=[0,1].}${\displaystyle \operatorname {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 ${\displaystyle (0,1),}${\displaystyle (0,1),} and so must have positive ${\displaystyle \mu }${\displaystyle \mu }-measure.

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. ^[2]

On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure ${\displaystyle 0.}${\displaystyle 0.} An example of this is given by adding the first uncountable ordinal ${\displaystyle \Omega }${\displaystyle \Omega } to the previous example: the support of the measure is the single point ${\displaystyle \Omega ,}${\displaystyle \Omega ,} which has measure ${\displaystyle 0.}${\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)
• Bogachev, V. I. (2007). Measure theory. Vol. 2. Springer Berlin Heidelberg. ISBN 978-3-540-34514-5.
• Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276. ISBN 0-8218-3889-X. MR 2169627 (See chapter 2, section 2)
• Teschl, Gerald (2009). Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS.(See chapter 3, section 2)

• Measures (measure theory)
• Measure theory

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