Definition:Convolution of Measures

Definition
Let $\mu$ and $\nu$ be measures on the Borel $\sigma$-algebra $\BB^n$ on $\R^n$.

The convolution of $\mu$ and $\nu$, denoted $\mu * \nu$, is the measure defined by:


 * $\ds \mu * \nu: \BB^n \to \overline \R, \map {\mu * \nu} B := \int \map {\chi_B} {x + y} \map {\rd \mu} x \map {\rd \nu} y$

where $\chi_B$ is the characteristic function of $B$.

Also known as
Some sources prefer the original German term Faltung (literally: folding) over convolution.

Also see

 * Definition:Convolution of Measurable Functions
 * Definition:Convolution of Measurable Function and Measure