Convolution of Measurable Functions is Bilinear

Theorem
Let $\BB^n$ be the Borel $\sigma$-algebra on $\R^n$.

Let $f, f', g, g': \R^n \to \R$ be $\BB^n$-measurable functions.

Then for all $\lambda \in \R$:


 * $\paren {\lambda f + f'} * g = \lambda \paren {f * g} + f' * g$
 * $f * \paren {\lambda g + g'} = \lambda \paren {f * g} + f * g'$

provided the convolutions in these expressions exist.

That is, convolution $*$ is a bilinear operation.