Definition:Convolution (Measure Theory)

This page is about Convolution in the context of Measure Theory. For other uses, see Convolution.

Definition

Let $\BB^n$ be the Borel $\sigma$-algebra on $\R^n$, and let $\lambda^n$ be Lebesgue measure on $\R^n$.

Let $f, g: \R^n \to \R$ be $\BB^n$-measurable functions such that for all $x \in \R^n$:

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

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

$\ds f * g: \R^n \to \R, \map {f * g} x := \int_{\R^n} \map f {x - y} \map g y \rd \map {\lambda^n} y$

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 $x \in \R^n$:

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

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

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

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 $B$.

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