Is there a "correct" or "best" way to define convolution of two (Borel) probability measures on $[0,1]$ to yield another probability measure on $[0,1]$?
Recall that the convolution, $\mu * \nu$, of $\mu$ and $\nu$ is the pushforward of the product measure, $\mu \otimes \nu$, via the map $(x,y) \mapsto x+y$. So in general, $\mu * \nu$ is a measure on $[0,2]$
It seems there are at least two reasonable approaches to get a sort of convolution $\mu\#\nu$ of probability measures on $[0,1]$ that is also a probability measure on $[0,1]$.
Approach 1: Define $\mu \# \nu$ as the pushforward of $\mu \otimes \nu$ via the map $(x,y) \mapsto x+ y \mod 1$.
Approach 2: Proceed as usual, obtaining $\mu*\nu$ a probability measure on $[0,2]$, then use the bijection $x \mapsto 2x$ between $[0,1]$ and $[0,2]$ to get a measure defined only on $[0,1]$; i.e. define $\mu \# \nu ([a,b]) = \mu*\nu([2a,2b])$.
What are the advantages and disadvantages to these approaches? Is there another approach that should be considered?