# Strictly positive measure

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In mathematics, strict positivity is a concept in measure theory. Intuitively, a **strictly positive measure** is one that is "nowhere zero", or that it is zero "only on points".

## 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

## 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,*δ*_{0}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*= {∅,**R**}, then*δ*_{0}is strictly positive. This example illustrates the importance of the topology in determining strict positivity. - Gaussian measure on Euclidean space
**R**^{n}(with its Borel topology and σ-algebra) is strictly positive.- Wiener measure on the space of continuous paths in
**R**^{n}is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.

- Wiener measure on the space of continuous paths in
- Lebesgue measure on
**R**^{n}(with its Borel topology and σ-algebra) is strictly positive. - The trivial measure is never strictly positive, regardless of the space
*X*or the topology used, except when*X*is empty.

## Properties

- If
*μ*and*ν*are two measures on a measurable topological space (X, Σ), with*μ*strictly positive and also absolutely continuous with respect to*ν*, then*ν*is strictly positive as well. The proof is simple: let*U*⊆*X*be an arbitrary open set; since*μ*is strictly positive,*μ*(*U*) > 0; by absolute continuity,*ν*(*U*) > 0 as well. - Hence, strict positivity is an invariant with respect to equivalence of measures.

## See also

- Support (measure theory): a measure is strictly positive if and only if its support is the whole space.