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 $\alpha$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.5 \ \text{(ii)}$