Strictly positive measure explained

In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points".

Definition

Let

be a Hausdorff topological space and let be a

\sigma

-algebra on that contains the topology (so that every open set is a measurable set, and is at least as fine as the Borel

\sigma

-algebra on ). Then a measure on is called strictly positive if every non-empty open subset of has strictly positive measure.

More concisely,

is strictly positive if and only if for all such that

Examples

• Counting measure on any set

(with any topology) is strictly positive.

• Dirac measure is usually not strictly positive unless the topology

is particularly "coarse" (contains "few" sets). For example, on the real line with its usual Borel topology and -algebra is not strictly positive; however, if is equipped with the trivial topology then is strictly positive. This example illustrates the importance of the topology in determining strict positivity. (with its Borel topology and -algebra) is strictly positive.

• • Wiener measure on the space of continuous paths in

is a strictly positive measure - Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.

• Lebesgue measure on

(with its Borel topology and -algebra) is strictly positive.

• The trivial measure is never strictly positive, regardless of the space

or the topology used, except when is empty.

Properties

• If

and are two measures on a measurable topological space with strictly positive and also absolutely continuous with respect to then is strictly positive as well. The proof is simple: let be an arbitrary open set; since is strictly positive, by absolute continuity, as well.

• Hence, strict positivity is an invariant with respect to equivalence of measures.

See also

• − a measure is strictly positive if and only if its support is the whole space.