Definition:Convolution of Measurable Function and Measure

Definition

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

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

$\ds \int_{\R^n} \map f {x - y} \rd \map \mu y$

is finite.

The convolution of $f$ and $\mu$ is the mapping $f * \mu: \R^n \to \R$ defined as:

$\ds \forall x \in \R^n: \map {f * \mu} x := \int_{\R^n} \map f {x - y} \rd \map \mu y$

Also known as

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

Also see

• Convolution of Measurable Functions
• Convolution of Measures

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.4$