Convolution of Measures as Pushforward Measure

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

Let $\alpha: \R^n \times \R^n \to \R^n, \ \map \alpha {\mathbf x, \mathbf y} = \mathbf x + \mathbf y$ be vector addition on $\R^n$.

Then we have the following equality of measures on $\BB^n$:


 * $\mu * \nu = \map {\alpha_*} {\mu \times \nu}$

where $\mu * \nu$ is the convolution of $\mu$ and $\nu$, and $\map {\alpha_*} {\mu \times \nu}$ is the pushforward of the product measure $\mu \times \nu$ under $\alpha$.