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

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

More concisely,   is strictly positive if and only if for all   such that  

Examples

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

Properties

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  • If   and   are two measures on a measurable topological space   with   strictly positive and also absolutely continuous with respect to   then   is strictly positive as well. The proof is simple: let   be an arbitrary open set; since   is strictly positive,   by absolute continuity,   as well.
  • Hence, strict positivity is an invariant with respect to equivalence of measures.

See also

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References

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