What is meant by "on se ramène par régularisation"?

Question

I am currently attempting to translate the paper 'Sur l'équation de convolution $\mu = \mu \ast \sigma$' by Choquet and Deny. In the paper, a locally compact abelian group $G$ and a positive measure $\sigma$ on $G$ are given and the measures $\mu$ which satisfy the convolution equation above are found. I've mostly been able to translate the paper, but there are a few phrases which I think are important which I don't understand.

The first phrase is

Inversement pour montrer que toute solution $\mu$ bornée est périodique on se ramène par régularisation au cas où $\mu$ est une fonction $f$ bornée uniformément continue sur $G$, et l'on peut supposer G dénombrable à l'infini.

In particular, the following part

$\mu$ bornée est périodique on se ramène par régularisation au cas où $\mu$ est une fonction $f$

What is meant there?

The second similar phrase I'm wondering about is

..., i.e. telles que les régularisées $\mu \ast \varphi$ soient des fonctions.

Finally, I'm also wondering about

donc telle que $g_0(x) = 2 \alpha$ en tout point $x$ du symétrique de $S_\sigma$ [support of $\sigma$] par rapport à l'origine O, done du semi-groupe engendré.

I'd greatly appreciate any help.

Answer 1

I am unsure what is meant by "régularisation". It might mean without loss of generality, in this case the first sentence translates to

Conversely, to show that any bounded solution $\mu$ is periodic, it suffices to show the case where $\mu$ is a uniformly continuous bounded function on $G$, where $G$ is assumed to be a countably infinite set.

As of the second sentence, you should provide the whole context. "Done" is not word in French, so your last sentence hardly makes sense. Perhaps you mean "donc", which translates to "thus"?