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

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

Examples

• Counting measure on any set ${\displaystyle X}$ (with any topology) is strictly positive.
• Dirac measure is usually not strictly positive unless the topology ${\displaystyle T}$ is particularly "coarse" (contains "few" sets). For example, ${\displaystyle \delta _{0}}$ on the real line ${\displaystyle \mathbb {R} }$ with its usual Borel topology and ${\displaystyle \sigma }$-algebra is not strictly positive; however, if ${\displaystyle \mathbb {R} }$ is equipped with the trivial topology ${\displaystyle T=\{\varnothing ,\mathbb {R} \},}$ then ${\displaystyle \delta _{0}}$ is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
• Gaussian measure on Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (with its Borel topology and ${\displaystyle \sigma }$-algebra) is strictly positive.
  • Wiener measure on the space of continuous paths in ${\displaystyle \mathbb {R} ^{n}}$ is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
• Lebesgue measure on ${\displaystyle \mathbb {R} ^{n}}$ (with its Borel topology and ${\displaystyle \sigma }$-algebra) is strictly positive.
• The trivial measure is never strictly positive, regardless of the space ${\displaystyle X}$ or the topology used, except when ${\displaystyle X}$ is empty.

Properties

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

See also

• Support (measure theory) – given a Borel measure, the set of those points whose neighbourhoods always have positive measurePages displaying wikidata descriptions as a fallback − a measure is strictly positive if and only if its support is the whole space.

References