Let $G$ be a topological group. Say that $A\subseteq G$ is totally bounded if for any neighborhood of $e$, $U$, there exists a finite subset $F\subseteq G$ such that $A\subseteq FU$. $G$ is locally totally bounded if there exists a neighborhood of $e$ that is totally bounded.
It seems to be a well-established fact that a locally totally bounded group may be densely extended to a unique locally compact group. That is, if $G$ is locally totally bounded, there exists some unique (up to isomorphism) locally compact group $\overline{G}$ such that $G$ is a dense subgroup of $\overline{G}$. Apparently, this was originally proved in Weil's Sur les espaces à structure uniforme et sur la topologie générale (1938), (chapter 3).
I can't find that book anywhere. Does anyone know the proof for this? Is it very complicated?
Here, (pg. 398) they seem to discuss this in terms of Raikov completions, but it seemed a bit complicated for me.