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| | The writer is known by the name of Figures Lint. California is where I've always been residing and I adore each working day residing here. Doing ceramics is what her family members and her enjoy. Hiring has been my occupation for some time but I've currently applied for an additional one.<br><br>My webpage [http://facehack.ir/index.php?do=/profile-110/info/ facehack.ir] |
| 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".
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| ==Definition==
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| Let (''X'', ''T'') be a [[Hausdorff space|Hausdorff]] [[topological space]] and let Σ be a [[sigma algebra|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 sigma algebra|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.
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| In more condensed notation, ''μ'' is strictly positive [[if and only if]]
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| :<math>\forall U \in T \mbox{ s.t. } U \neq \emptyset, \mu (U) > 0.</math>
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| ==Examples==
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| * [[Counting measure]] on any set ''X'' (with any topology) is strictly positive.
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| * [[Dirac measure]] is usually not strictly positive unless the topology ''T'' is particularly "coarse" (contains "few" sets). For example, ''δ''<sub>0</sub> 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''<sub>0</sub> is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
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| * [[Gaussian measure]] on [[Euclidean space]] '''R'''<sup>''n''</sup> (with its Borel topology and σ-algebra) is strictly positive.
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| ** [[Wiener measure]] on the space of continuous paths in '''R'''<sup>''n''</sup> is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
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| * [[Lebesgue measure]] on '''R'''<sup>''n''</sup> (with its Borel topology and σ-algebra) is strictly positive.
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| * The [[trivial measure]] is never strictly positive, regardless of the space ''X'' or the topology used, except when ''X'' is empty.
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| ==Properties==
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| * 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.
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| * Hence, strict positivity is an [[invariant (mathematics)|invariant]] with respect to [[equivalence of measures]].
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| ==See also==
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| * [[Support (measure theory)]]: a measure is strictly positive [[if and only if]] its support is the whole space.
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| {{DEFAULTSORT:Strictly Positive Measure}}
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| [[Category:Measures (measure theory)]]
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The writer is known by the name of Figures Lint. California is where I've always been residing and I adore each working day residing here. Doing ceramics is what her family members and her enjoy. Hiring has been my occupation for some time but I've currently applied for an additional one.
My webpage facehack.ir