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

Let [math]\displaystyle{ (X, T) }[/math] be a Hausdorff topological space and let [math]\displaystyle{ \Sigma }[/math] be a [math]\displaystyle{ \sigma }[/math]-algebra on [math]\displaystyle{ X }[/math] that contains the topology [math]\displaystyle{ T }[/math] (so that every open set is a measurable set, and [math]\displaystyle{ \Sigma }[/math] is at least as fine as the Borel [math]\displaystyle{ \sigma }[/math]-algebra on [math]\displaystyle{ X }[/math]). Then a measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (X, \Sigma) }[/math] is called strictly positive if every non-empty open subset of [math]\displaystyle{ X }[/math] has strictly positive measure.

More concisely, [math]\displaystyle{ \mu }[/math] is strictly positive if and only if for all [math]\displaystyle{ U \in T }[/math] such that [math]\displaystyle{ U \neq \varnothing, \mu (U) \gt 0. }[/math]

Examples

• Counting measure on any set [math]\displaystyle{ X }[/math] (with any topology) is strictly positive.
• Dirac measure is usually not strictly positive unless the topology [math]\displaystyle{ T }[/math] is particularly "coarse" (contains "few" sets). For example, [math]\displaystyle{ \delta_0 }[/math] on the real line [math]\displaystyle{ \R }[/math] with its usual Borel topology and [math]\displaystyle{ \sigma }[/math]-algebra is not strictly positive; however, if [math]\displaystyle{ \R }[/math] is equipped with the trivial topology [math]\displaystyle{ T = \{\varnothing, \R\}, }[/math] then [math]\displaystyle{ \delta_0 }[/math] is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
• Gaussian measure on Euclidean space [math]\displaystyle{ \R^n }[/math] (with its Borel topology and [math]\displaystyle{ \sigma }[/math]-algebra) is strictly positive.
  • Wiener measure on the space of continuous paths in [math]\displaystyle{ \R^n }[/math] is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
• Lebesgue measure on [math]\displaystyle{ \R^n }[/math] (with its Borel topology and [math]\displaystyle{ \sigma }[/math]-algebra) is strictly positive.
• The trivial measure is never strictly positive, regardless of the space [math]\displaystyle{ X }[/math] or the topology used, except when [math]\displaystyle{ X }[/math] is empty.

Properties

• If [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] are two measures on a measurable topological space [math]\displaystyle{ (X, \Sigma), }[/math] with [math]\displaystyle{ \mu }[/math] strictly positive and also absolutely continuous with respect to [math]\displaystyle{ \nu, }[/math] then [math]\displaystyle{ \nu }[/math] is strictly positive as well. The proof is simple: let [math]\displaystyle{ U \subseteq X }[/math] be an arbitrary open set; since [math]\displaystyle{ \mu }[/math] is strictly positive, [math]\displaystyle{ \mu(U) \gt 0; }[/math] by absolute continuity, [math]\displaystyle{ \nu(U) \gt 0 }[/math] 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.