Inmathematics, 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".
Let be a Hausdorfftopological space and let be a -algebraon that contains the topology (so that every open set is a measurable set, and is at least as fine as the Borel -algebraon). 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
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.
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 measureon (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.
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.