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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
for any Borel subset A of X. 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.
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