Convolution of Measures is Bilinear

Theorem
Let $\mu, \mu', \nu$ and $\nu'$ be measures on the Borel $\sigma$-algebra $\BB^n$ on $\R^n$.

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


 * $\paren {\lambda \mu + \mu'} * \nu = \lambda \paren {\mu * \nu} + \mu' * \nu$
 * $\mu * \paren {\lambda \nu + \nu'} = \lambda \paren {\mu * \nu} + \mu * \nu'$

where $*$ denotes convolution.

That is, convolution is a bilinear operation.