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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".

Let be a Hausdorff topological space and let be a -algebra on that contains the topology (so that every open set is a measurable set, and is at least as fine as the Borel -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

- 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.
- Gaussian measure on Euclidean space (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.

- 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.