Convolution of Measures as Pushforward Measure

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Theorem

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

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


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

$\mu * \nu = \alpha_* \left({\mu \times \nu}\right)$

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


Proof


Sources