It is defined to be the largest (closed) subset of
for which every open neighbourhood of every point of the set has positive measure.
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
Consider two examples: In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section: However, the idea of "local strict positivity" is not too far from a workable definition.
Some authors prefer to take the closure of the above set.
An equivalent definition of support is as the largest
(with respect to inclusion) such that every open set which has non-empty intersection with
This definition can be extended to signed and complex measures.
Use the Hahn decomposition theorem to write
is defined to be the union of the supports of its real and imaginary parts.
since it is an open set, has positive measure; hence,
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,
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.
is a Radon measure, a Borel set
is open, but it is not true in general: it fails if there exists a point
Thus, one does not need to "integrate outside the support": for any measurable function
The concept of support of a measure and that of spectrum of a self-adjoint linear operator on a Hilbert space are closely related.
is a regular Borel measure on the line
and its spectrum coincides with the essential range of the identity function
on the real line is a Dirac measure
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 (or 1) contains an open interval about 0 (or 1), which must intersect
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.
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