Support (measure theory)

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 ${\displaystyle \Sigma }$:

${\displaystyle \operatorname {supp} (\mu ):=\bigcap \{A={\overline {A}}\in \Sigma \mid \mu (A^{c})=0\},}$

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 ${\displaystyle \Sigma }$. 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 ${\displaystyle X\setminus \{x\in X\mid \mu (\{x\})=0\}}$. 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:

${\displaystyle \{x\in X\mid {\text{for some open }}N_{x}\ni x,\mu (N_{x})>0\}}$

(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 (X, T) be a topological space; let B(T) 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, B(T)). Then the support (or spectrum) of μ is defined as the set of all points x in X for which every open neighbourhood N_x of x has positive measure:

${\displaystyle \operatorname {supp} (\mu ):=\{x\in X\mid x\in N_{x}\in T\implies \mu (N_{x})>0\}.}$

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 C ∈ B(T) (with respect to inclusion) such that every open set which has non-empty intersection with C has positive measure, i.e. the largest C such that:

${\displaystyle (\forall U\in T)(U\cap C\neq \varnothing \implies \mu (U\cap C)>0).}$

• ${\displaystyle \operatorname {supp} (\mu _{1}+\mu _{2})=\operatorname {supp} (\mu _{1})\cup \operatorname {supp} (\mu _{2})}$
• 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 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, μ is strictly positive.
• The support of a measure is closed in X as its complement is the union of the open sets of measure 0.
• In general the support of a nonzero measure may be empty: see the examples below. However, if X is a topological Hausdorff space and μ is a Radon measure, a measurable set A outside the support has measure zero:

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

The converse is true if A is open, but it is not true in general: it fails if there exists a point x ∈ supp(μ) such that μ({x}) = 0 (e.g. Lebesgue measure).

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

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

• 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 ${\displaystyle \mu }$ 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

${\displaystyle D(A)=\{f\in L^{2}(\mathbb {R} ,d\mu )\mid xf(x)\in L^{2}(\mathbb {R} ,d\mu )\}}$

and its spectrum coincides with the essential range of the identity function ${\displaystyle x\mapsto x}$, which is precisely the support of ${\displaystyle \mu }$.[1]

Suppose that μ : Σ → [−∞, +∞] is a signed measure. Use the Hahn decomposition theorem to write

${\displaystyle \mu =\mu ^{+}-\mu ^{-},}$

where μ^± are both non-negative measures. Then the support of μ is defined to be

${\displaystyle \operatorname {supp} (\mu ):=\operatorname {supp} (\mu ^{+})\cup \operatorname {supp} (\mu ^{-}).}$

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.

• 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.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)