Strictly positive measure

Definition

Examples

• Counting measure on any set X is strictly positive.
• Dirac measure is usually not strictly positive unless the topology T is particularly "coarse". 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 = , then δ₀ is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
• Gaussian measure on Euclidean space Rⁿ 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 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, 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 > 0; by absolute continuity, ν > 0 as well.
• Hence, strict positivity is an invariant with respect to equivalence of measures.