Definition:Convolution (Measure Theory)

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Definition

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


Convolution of Measurable Functions

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

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

is finite.


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

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


Convolution of Measurable Function and Measure

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:

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


Convolution of Measures

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.