When is convolution associative?

Question

Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions.

What about the convolution of measures on an unimodular group $G$?

Answer 1

Convolution of finite Borel measures on a topological group is always well-defined and associative.

Answer 2

Not sure if it's the most general case but according to Kallenberg's Foundations of Modern Probability, convolution of measures (I suspect the measures need to be at least $\sigma$-finite here, it's not specified but the proof states that you should use Fubini's) on a measurable group (group endowed with a $\sigma$-algebra such that the group operations are measurable) is associative, so it's a bit more general than the answer above.