## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

## Convolution of mea

Let G be a topological group. If μ and ν are finite Borel measures on a group G, then their convolution μ∗ν is defined by

$(mu * u)(E) = int!!!int 1_E(xy) ,dmu(x) ,d u(y)$

for each measurable subset E of G. The convolution is also a finite measure, whose total variation satisfies

$|mu * u| le |mu| | u|. ,$

In the case when G is locally compact with (left-)Haar measure λ, and μ and ν are absolutely continuous with respect to a λ, so that each has a density function, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions.

If μ and ν are probability measures, then the convolution μ∗ν is the probability distribution of the sum X + Y of two independent random variables X and Y whose respective distributions are μ and ν.

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