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| In [[mathematics]] — specifically, in [[measure theory]] — '''Malliavin's absolute continuity lemma''' is a result due to the [[France|French]] [[mathematician]] [[Paul Malliavin]] that plays a foundational rôle in the regularity ([[smooth function|smoothness]]) [[theorem]]s of the [[Malliavin calculus]]. Malliavin's lemma gives a sufficient condition for a [[finite measure|finite]] [[Borel measure]] to be [[absolute continuity|absolutely continuous]] with respect to [[Lebesgue measure]].
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| ==Statement of the lemma== | |
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| Let ''μ'' be a finite Borel measure on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>. Suppose that, for every ''x'' ∈ '''R'''<sup>''n''</sup>, there exists a constant ''C'' = ''C''(''x'') such that
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| :<math>\left| \int_{\mathbf{R}^{n}} \mathrm{D} \varphi (y) (x) \, \mathrm{d} \mu(y) \right| \leq C(x) \| \varphi \|_{\infty}</math>
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| for every ''C''<sup>∞</sup> function ''φ'' : '''R'''<sup>''n''</sup> → '''R''' with [[compact support]]. Then ''μ'' is absolutely continuous with respect to ''n''-dimensional Lebesgue measure ''λ''<sup>''n''</sup> on '''R'''<sup>''n''</sup>. In the above, D''φ''(''y'') denotes the [[Fréchet derivative]] of ''φ'' at ''y'' and ||''φ''||<sub>∞</sub> denotes the [[supremum norm]] of ''φ''.
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| ==References==
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| * {{cite book
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| | last = Bell
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| | first = Denis R.
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| | title = The Malliavin calculus
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| | publisher = Dover Publications Inc.
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| | location = Mineola, NY
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| | year = 2006
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| | pages = x+113
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| | isbn = 0-486-44994-7
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| }} {{MathSciNet|id=2250060}} (See section 1.3)
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| * {{cite book
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| | last = Malliavin
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| | first = Paul
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| | authorlink = Paul Malliavin
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| | chapter = Stochastic calculus of variations and hypoelliptic operators
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| | title = Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976)
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| | pages= 195–263
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| | publisher = Wiley
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| | location = New York
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| | year = 1978
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| }} {{MathSciNet|id=536013}}
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| [[Category:Lemmas]]
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| [[Category:Measure theory]]
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Latest revision as of 18:33, 12 May 2014
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