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 $(X, T)$ be a Hausdorff topological space and let $\Sigma$ be a $\sigma$-algebra on $X$ that contains the topology $T$ (so that every open set is a measurable set, and $\Sigma$ is at least as fine as the Borel $\sigma$-algebra on $X$). Then a measure $\mu$ on $(X, \Sigma)$ is called strictly positive if every non-empty open subset of $X$ has strictly positive measure.

More concisely, $\mu$ is strictly positive if and only if for all $U \in T$ such that $U \neq \varnothing, \mu (U) > 0.$

Examples

* Counting measure on any set $X$ (with any topology) is strictly positive.
* Dirac measure is usually not strictly positive unless the topology $T$ is particularly "coarse" (contains "few" sets). For example, $\delta_0$ on the real line $\R$ with its usual Borel topology and $\sigma$-algebra is not strictly positive; however, if $\R$ is equipped with the trivial topology $T = \,$ then $\delta_0$ is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
* Gaussian measure on Euclidean space $\R^n$ (with its Borel topology and $\sigma$-algebra) is strictly positive.
** Wiener measure on the space of continuous paths in $\R^n$ is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
* Lebesgue measure on $\R^n$ (with its Borel topology and $\sigma$-algebra) is strictly positive.
* The trivial measure is never strictly positive, regardless of the space $X$ or the topology used, except when $X$ is empty.

Properties

* If $\mu$ and $\nu$ are two measures on a measurable topological space $(X, \Sigma),$ with $\mu$ strictly positive and also absolutely continuous with respect to $\nu,$ then $\nu$ is strictly positive as well. The proof is simple: let $U \subseteq X$ be an arbitrary open set; since $\mu$ is strictly positive, $\mu(U) > 0;$ by absolute continuity, $\nu(U) > 0$ 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.

References

{{DEFAULTSORT:Strictly Positive Measure

Measures (measure theory)