Strictly positive measure
- Strictly positive measure
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure 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
:
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, "δ"0 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 "δ"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 σ-algebra) is strictly positive.
** Wiener measure on the space of continuous paths in R"n" is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
* Lebesgue measure on R"n" (with its Borel topology and σ-algebra) is strictly positive.
* The trivial measure is never strictly positive, regardless of the space or topology used.
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.
ee also
* Support (measure theory): a measure is strictly positive if and only if its support is the whole space.
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