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| | Wilber Berryhill is what his spouse enjoys to call him and he completely enjoys this title. Office supervising is exactly where her primary income comes from. She is truly fond of caving but she doesn't have the time recently. I've always loved living in Alaska.<br><br>My website [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ psychic readings online] |
| In [[mathematics]], a '''quasi-invariant measure''' ''μ'' with respect to a transformation ''T'', from a [[measure space]] ''X'' to itself, is a [[measure (mathematics)|measure]] which, roughly speaking, is multiplied by a [[numerical function]] of ''T''. An important class of examples occurs when ''X'' is a [[smooth manifold]] ''M'', ''T'' is a [[diffeomorphism]] of ''M'', and ''μ'' is any measure that locally is a [[measure with base]] the [[Lebesgue measure]] on [[Euclidean space]]. Then the effect of ''T'' on μ is locally expressible as multiplication by the [[Jacobian matrix and determinant|Jacobian]] determinant of the derivative ([[pushforward (differential)|pushforward]]) of ''T''.
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| To express this idea more formally in [[measure theory]] terms, the idea is that the [[Radon–Nikodym derivative]] of the transformed measure μ′ with respect to ''μ'' should exist everywhere; or that the two measures should be [[Equivalence (measure theory)|equivalent]] (i.e. mutually [[absolutely continuous]]):
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| :<math>\mu' = T_{*} (\mu) \approx \mu.</math>
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| That means, in other words, that ''T'' preserves the concept of a set of [[measure zero]]. Considering the whole equivalence class of measures ''ν'', equivalent to ''μ'', it is also the same to say that ''T'' preserves the class as a whole, mapping any such measure to another such. Therefore the concept of quasi-invariant measure is the same as ''invariant measure class''.
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| In general, the 'freedom' of moving within a measure class by multiplication gives rise to [[Oseledets theorem#Cocycles|cocycles]], when transformations are composed.
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| As an example, [[Gaussian measure]] on [[Euclidean space]] '''R'''<sup>''n''</sup> is not invariant under translation (like Lebesgue measure is), but is quasi-invariant under all translations.
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| It can be shown that if ''E'' is a [[separable space|separable]] [[Banach space]] and ''μ'' is a [[Locally finite measure|locally finite]] [[Borel measure]] on ''E'' that is quasi-invariant under all translations by elements of ''E'', then either dim(''E'') < +∞ or ''μ'' is the [[trivial measure]] ''μ'' ≡ 0.
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| {{DEFAULTSORT:Quasi-Invariant Measure}}
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| [[Category:Measures (measure theory)]]
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| [[Category:Dynamical systems]]
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