Definition:Convolution of Measures

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

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


 * $\displaystyle \mu * \nu: \mathcal B^n \to \overline \R, \mu * \nu \left({B}\right) := \int_{\R^n} \chi_B \left({x + y}\right) \, \mathrm d \mu \left({x}\right) \, \mathrm d \nu \left({y}\right)$

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