This paper continues
work initiated in Part I. The central question is one of characterizing a net
{Aα} of Borel sets in the group G which averages a so-called regular set
function S on G in the sense that λ(Aα)−1S(Aα) has a limit (depending only
on S), where λ is Haar measure. In Part I a sufficient condition for {Aα}
to always average was derived; here we show that a “natural” relaxation
of this condition is no longer sufficient for all regular S, at the same time
essentially characterizing those S which may still be averaged. Moreover,
the role of Følner summing sequences is considered in this context. Finally,
properties of regular set functions are derived which may be of independent
interest.
