Strictly positive measure
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 -algebra on that contains the topology . 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 is strictly positive.
- Dirac measure is usually not strictly positive unless the topology is particularly "coarse". 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.
- Gaussian measure on Euclidean space 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 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.
- Any uniformly distributed measure on a metric space is strictly positive. Because if there is an nonempty open set with zero measure, then the measure of some open balls will be zero, which contradicts the definition of uniformly distributed'''.'''