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 $\mathbb {R}$ with its usual Borel topology and $\sigma$-algebra is not strictly positive; however, if $\mathbb {R}$ is equipped with the trivial topology $T=\{\varnothing ,\mathbb {R} \},$ then $\delta _{0}$ is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
• Gaussian measure on Euclidean space $\mathbb {R} ^{n}$ (with its Borel topology and $\sigma$-algebra) is strictly positive.
  • Wiener measure on the space of continuous paths in $\mathbb {R} ^{n}$ is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
• Lebesgue measure on $\mathbb {R} ^{n}$ (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

• Support (measure theory) – Concept in mathematics − a measure is strictly positive if and only if its support is the whole space.