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 <math>(\Omega, \mathcal{T})</math> be a topological space, and let <math>\mathcal{F}</math> be a sigma algebra on <math>\Omega</math> that contains the topology <math>\mathcal{T}</math>, so every open set in <math>\Omega</math> is measurable. Then a measure <math>\mu : \mathcal{F} \to [0, + \infty]</math> on <math>\Omega</math> is called strictly positive if every non-empty open set <math>\varnothing \neq U \in \mathcal{T}</math> has positive measure <math>\mu (U) > 0</math>.

[edit] Examples

Dirac measure is usually not strictly positive unless the topology <math>\mathcal{T}</math> is particularly "coarse" (contains "few" sets). For example, <math>\delta_{0}</math> on <math>(\mathbb{R}, \mathrm{Borel} (\mathbb{R}))</math> is not strictly positive; however, if we give <math>\mathbb{R}</math> the trivial topology <math>\mathcal{T} = \{ \varnothing, \mathbb{R} \}</math>, then <math>\delta_{0}</math> is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
Counting measure on any set <math>\Omega</math> (with any topology) is strictly positive.
Gaussian measure on Euclidean space <math>\mathbb{R}^{n}</math> (with its Borel topology and sigma algebra) is strictly positive.
Lebesgue measure on <math>\mathbb{R}^{n}</math> (with its Borel topology and sigma algebra) is strictly positive.