Definition:Convolution of Measurable Function and Measure

Definition
Let $\mu$ be a measure on the Borel $\sigma$-algebra $\mathcal B^n$ on $\R^n$.

Let $f: \R^n \to \R$ be a $\mathcal B^n$-measurable function such that for all $x \in \R^n$:


 * $\displaystyle \int_{\R^n} f \left({x - y}\right) \, \mathrm d \mu \left({y}\right)$

is finite.

The convolution of $f$ and $\mu$, denoted $f * \mu$, is the mapping defined by:


 * $\displaystyle f * \mu: \R^n \to \R, f * \mu \left({x}\right) := \int_{\R^n} f \left({x - y}\right) \, \mathrm d \mu \left({y}\right)$

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

Also see

 * Convolution of Measurable Functions
 * Convolution of Measures