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| In [[mathematics]], an '''idempotent measure''' on a [[metric group]] is a [[probability measure]] that equals its [[convolution]] with itself; in other words, an idempotent measure is an [[idempotent element]] in the [[topological semigroup]] of probability measures on the given metric group.
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| Explicitly, given a metric group ''X'' and two probability measures ''μ'' and ''ν'' on ''X'', the convolution ''μ'' ∗ ''ν'' of ''μ'' and ''ν'' is the measure given by
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| :<math>(\mu * \nu) (A) = \int_{X} \mu (A x^{-1}) \, \mathrm{d} \nu (x) = \int_{X} \nu (x^{-1} A) \, \mathrm{d} \mu (x)</math>
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| for any Borel subset ''A'' of ''X''. (The equality of the two integrals follows from [[Fubini's theorem]].) With respect to the topology of [[weak convergence of measures]], the operation of convolution makes the space of probability measures on ''X'' into a topological semigroup. Thus, ''μ'' is said to be an idempotent measure if ''μ'' ∗ ''μ'' = ''μ''.
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| It can be shown that the only idempotent probability measures on a [[complete space|complete]], [[separable space|separable]] metric group are the normalized [[Haar measure]]s of [[compact space|compact]] [[subgroup]]s.
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| ==References==
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| * {{cite book
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| | last = Parthasarathy
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| | first = K. R.
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| | title = Probability measures on metric spaces
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| |publisher = AMS Chelsea Publishing, Providence, RI
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| | year = 2005
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| | pages = pp.xii+276
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| | isbn = 0-8218-3889-X
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| }} {{MathSciNet|id=2169627}} (See chapter 3, section 3.)
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| [[Category:Group theory]]
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| [[Category:Measures (measure theory)]]
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| [[Category:Metric geometry]]
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Latest revision as of 12:05, 5 June 2014
My name is Saundra and I am studying Continuing Education and Summer Sessions and Modern Languages and Classics at Porto Alegre / Brazil.
my web page wordpress backup plugin