Definition:Convolution (Measure Theory)

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

(f*g)(x) = \int_{\R^N} f(y) g(x-y) \,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.