Strictly positive measure
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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".
[edit] 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
[edit] 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
- 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.
[edit] See also
- Support (measure theory): a measure is strictly positive if and only if its support is the whole space.
