Definition:Convolution of Measures
Jump to navigation
Jump to search
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
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.4$