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".

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

More concisely, ${\displaystyle \mu }${\displaystyle \mu } is strictly positive if and only if for all ${\displaystyle U\in T}${\displaystyle U\in T} such that ${\displaystyle U\neq \varnothing ,\mu (U)>0.}${\displaystyle U\neq \varnothing ,\mu (U)>0.}

• Counting measure on any set ${\displaystyle X}${\displaystyle X} (with any topology) is strictly positive.
• Dirac measure is usually not strictly positive unless the topology ${\displaystyle T}${\displaystyle T} is particularly "coarse" (contains "few" sets). For example, ${\displaystyle \delta _{0}}${\displaystyle \delta _{0}} on the real line ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } with its usual Borel topology and ${\displaystyle \sigma }${\displaystyle \sigma }-algebra is not strictly positive; however, if ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } is equipped with the trivial topology ${\displaystyle T=\{\varnothing ,\mathbb {R} \},}${\displaystyle T=\{\varnothing ,\mathbb {R} \},} then ${\displaystyle \delta _{0}}${\displaystyle \delta _{0}} is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
• Gaussian measure on Euclidean space ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} (with its Borel topology and ${\displaystyle \sigma }${\displaystyle \sigma }-algebra) is strictly positive.
  • Wiener measure on the space of continuous paths in ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
• Lebesgue measure on ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} (with its Borel topology and ${\displaystyle \sigma }${\displaystyle \sigma }-algebra) is strictly positive.
• The trivial measure is never strictly positive, regardless of the space ${\displaystyle X}${\displaystyle X} or the topology used, except when ${\displaystyle X}${\displaystyle X} is empty.

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

• Support (measure theory) – Concept in mathematics − a measure is strictly positive if and only if its support is the whole space.

• Measures (measure theory)

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