Definition:Convolution (Measure Theory)

Definition
The convolution of two Lebesgue integrable functions $f, g: \R^N \to \C$ is the function $f * g: \R^N \to \C$ defined almost everywhere by:
 * $\displaystyle \left({f * g}\right) \left({x}\right) = \int_{\R^N} f \left({y}\right) g \left({x-y}\right) \ dy \quad \text{ a.e. } x \in \R^N$

Related definitions
One may also define the convolution of two finite measures on $\R^N$, and in general the convolution of a distribution with a distribution of compact support.