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

[edit] Examples

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