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 $$T = \{\varnothing, \R\},$$ 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 &mdash; 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.