Convolution of Measurable Functions is Bilinear

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Theorem

Let $\mathcal B^n$ be the Borel $\sigma$-algebra on $\R^n$.

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


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

$\left({\lambda f + f'}\right) * g = \lambda \left({f * g}\right) + f' * g$
$f * \left({\lambda g + g'}\right) = \lambda \left({f * g}\right) + f * g'$

provided the convolutions in these expressions exist.

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


Proof


Sources