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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. | |||
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 | |||
:<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> | |||
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''. | |||
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. | |||
==References== | |||
* {{cite book | |||
| last = Parthasarathy | |||
| first = K. R. | |||
| title = Probability measures on metric spaces | |||
|publisher = AMS Chelsea Publishing, Providence, RI | |||
| year = 2005 | |||
| pages = pp.xii+276 | |||
| isbn = 0-8218-3889-X | |||
}} {{MathSciNet|id=2169627}} (See chapter 3, section 3.) | |||
[[Category:Group theory]] | |||
[[Category:Measures (measure theory)]] | |||
[[Category:Metric geometry]] | |||
Revision as of 20:55, 7 November 2013
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.
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
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.
It can be shown that the only idempotent probability measures on a complete, separable metric group are the normalized Haar measures of compact subgroups.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet (See chapter 3, section 3.)