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 it is zero "only on points".

Definition

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

In more condensed notation, μ is strictly positive if and only if

$\forall U \in T \mbox{ s.t. } U \neq \emptyset, \mu (U) > 0.$

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, δ₀ on the real line R with its usual Borel topology and σ-algebra is not strictly positive; however, if R is equipped with the trivial topology T = {∅, R}, then δ₀ is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
• Gaussian measure on Euclidean space Rⁿ (with its Borel topology and σ-algebra) is strictly positive.
  • Wiener measure on the space of continuous paths in Rⁿ is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
• Lebesgue measure on Rⁿ (with its Borel topology and σ-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 μ and ν are two measures on a measurable topological space (X, Σ), with μ strictly positive and also absolutely continuous with respect to ν, then ν is strictly positive as well. The proof is simple: let U ⊆ X be an arbitrary open set; since μ is strictly positive, μ(U) > 0; by absolute continuity, ν(U) > 0 as well.
• Hence, strict positivity is an invariant with respect to equivalence of measures.

See also

• Support (measure theory): a measure is strictly positive if and only if its support is the whole space.