  We first give simplified and corrected accounts of some results in
\cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use
a classical theorem of Turing \cite{Turing} to give a simplified proof that any
definable compactification of a pseudofinite group has an abelian connected
component. We then discuss the relationship between Turing's work, the
Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan
\cite{Kazh} on approximate homomorphisms, and we use this to widen our scope
from finite groups to amenable groups. In particular, we develop a suitable
continuous logic framework for dealing with definable homomorphisms from
pseudo-amenable groups to compact Lie groups. Together with the stabilizer
theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative)
analogue of Bogolyubov's Lemma for sets of positive measure in discrete
amenable groups. We conclude with brief remarks on the case of amenable
topological groups.
