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{{one source |date=March 2024}}
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 ''&mu;'' and ''&nu;'' on ''X'', the convolution ''&mu;''&nbsp;&lowast;&nbsp;''&nu;'' of ''&mu;'' and ''&nu;'' 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, ''&mu;'' is said to be an idempotent measure if ''&mu;''&nbsp;&lowast;&nbsp;''&mu;''&nbsp;=&nbsp;''&mu;''.

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==
{{refbegin}}
* {{cite book
|     last = Parthasarathy
|    first = K. R.
|    title = Probability measures on metric spaces
|publisher = AMS Chelsea Publishing, Providence, RI
|     year = 2005
|    pages = xii+276
|     isbn = 0-8218-3889-X
}} {{MathSciNet|id=2169627}} (See chapter 3, section 3.)
{{refend}}

[[Category:Group theory]]
[[Category:Measures (measure theory)]]
[[Category:Metric geometry]]


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