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

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

[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
T = \{ \emptyset, \mathbb{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.

[edit] See also

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