\begin{definition}[Definition:Convolution of Measurable Functions]
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 $x \in \R^n$:
:$\ds \int_{\R^n} \map f {x - y} \map g y \rd \map {\lambda^n} y$
is finite.
The '''convolution of $f$ and $g$''', denoted $f * g$, is the mapping defined by:
:$\ds f * g: \R^n \to \R, \map {f * g} x := \int_{\R^n} \map f {x - y} \map g y \rd \map {\lambda^n} y$
\end{definition}