Support (measure theory)

In mathematics, the support of a measure on a measurable topological space is a precise notion of where in the space the measure "lives". It is defined to be the largest subset of for which every open neighbourhood of every point of the set has positive measure.

Motivation

A measure on a measurable space is really a function Therefore, in terms of the usual definition of support, the support of is a subset of the σ-algebra
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 What we really want to know is where in the space the measure is non-zero. Consider two examples:

Lebesgue measure on the real line It seems clear that "lives on" the whole of the real line.
A Dirac measure at some point Again, intuition suggests that the measure "lives at" the point 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:

We could remove the points where is zero, and take the support to be the remainder This might work for the Dirac measure but it would definitely not work for since the Lebesgue measure of any singleton is zero, this definition would give empty support.
By comparison with the notion of [Strictly positive number|positive measure|strict positivity] of measures, we could take the support to be the set of all points with a neighbourhood of positive measure: . It is also too simplistic: by taking for all points this would make the support of every measure except the zero measure the whole of

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

Definition

Let be a topological space; let denote the Borel σ-algebra on i.e. the smallest sigma algebra on that contains all open sets Let be a measure on. Then the support of is defined as the set of all points in for which every open neighbourhood of 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 such that every open set which has non-empty intersection with has positive measure, i.e. the largest such that:

Signed and complex measures

This definition can be extended to signed and complex measures.
Suppose that is a signed measure. Use the Hahn decomposition theorem to write
where are both non-negative measures. Then the support of is defined to be
Similarly, if is a complex measure, the support of is defined to be the union of the supports of its real and imaginary parts.

Properties

holds.
A measure on is strictly positive if and only if it has support If is strictly positive and is arbitrary, then any open neighbourhood of since it is an open set, has positive measure; hence, so Conversely, if then every non-empty open set has positive measure; hence, is strictly positive.
The support of a measure is closed in as its complement is the union of the open sets of measure
In general the support of a nonzero measure may be empty: see the examples below. However, if is a Hausdorff topological space and is a Radon measure, a Borel set outside the support has measure zero:
The converse is true if is open, but it is not true in general: it fails if there exists a point such that . Thus, one does not need to "integrate outside the support": for any measurable function or
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 is a regular Borel measure on the line then the multiplication operator is self-adjoint on its natural domain
and its spectrum coincides with the essential range of the identity function which is precisely the support of

Examples

Lebesgue measure

In the case of Lebesgue measure on the real line consider an arbitrary point Then any open neighbourhood of must contain some open interval for some This interval has Lebesgue measure so Since was arbitrary,

Dirac measure

In the case of Dirac measure let and consider two cases:

if then every open neighbourhood of contains so
on the other hand, if then there exists a sufficiently small open ball around that does not contain so

We conclude that is the closure of the singleton set which is itself.
In fact, a measure on the real line is a Dirac measure for some point if and only if the support of is the singleton set Consequently, Dirac measure on the real line is the unique measure with zero variance.

A uniform distribution

Consider the measure on the real line defined by
i.e. a uniform measure on the open interval A similar argument to the Dirac measure example shows that Note that the boundary points 0 and 1 lie in the support: any open set containing 0 contains an open interval about 0, which must intersect and so must have positive -measure.

A nontrivial measure whose support is empty

The space of all countable ordinals with the topology generated by "open intervals" is a locally compact Hausdorff space. The 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

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