The task: Let $T:\mathbb R^d \rightarrow \mathbb R^d$ be a linear map. For $\mu, \nu \in M^1(\mathbb R^d)$ (so $\mu,\nu$ are probability measures), I have to show:
$T(\mu\ast\nu)= T(\mu)\ast T(\nu)$
So we have defined the convolution of measures as $\mu \ast \nu:=A(\mu \times\nu)$, with $A: \mathbb R^d \times \mathbb R^d \rightarrow \mathbb R^d$, $(x,y)\mapsto x+y$
So I have: $T(\mu)\ast T(\nu)=A(T(\mu) \times T(\nu))$ and
$T(\mu\ast\nu)=T(A(\mu \times\nu))=A(T(\mu \times \nu))=A(T(\mu) \times T(\nu))$, because $A,T$ are linear transformations. Is it okay that way, or am I doing something wrong?