Convolution of measures and one application

If $\mu,\nu$ are two finite Borel measures on $R^d,$ the their convolution is the push-down of $\mu\times \nu$ under the addition map $(x,y)\to x+y,$ that is 

$$(\mu\ast \nu)(A)=\int\int I_A(x+y)d\mu(x)d\nu(y)$$ for any Borel set $A.$

Note that the addition map is continuous and hence measurable,  $(\mu\ast \nu)(\cdot)$ is a measure on $R^d \times R^d.$ 

Remark: The self-convolution $\mu\ast \mu$ of a singular measure on $R$ can be absolutely continuous, and the density can be H\"{o}lder continuous. 

The following is a nice application.

 Let $A\subset R^d$ be a Salme set. Then for any Borel set $B\subset R^d,$

$$\dim_H(A+B)=\min\{\dim_H A+\dim_H B, d\}$$.

Moreover, if $\dim_H A+\dim_H B>d$, then $A+B$ has positive Lebesgue measure.

For the proof of the above result, see "A class of random Cantor measures, with applications" by P. Shmerkin and V. Suomala.