In 1967 the first author and
Karl Stromberg published a theorem concerning generalized limits of Riemann sums
on locally compact groups. The setting is a locally compact group G and an
increasing sequence Hn of closed subgroups whose union is dense in G. The theorem
was shown to hold provided that the restriction of the modular function on G to Hn
agrees with the modular function of Hn for all large n. This hypothesis holds
in many cases and, in fact, Ross and Stromberg were unable to determine
whether the hypothesis was really needed for the theorem or even whether this
hypothesis always holds. An example is provided which shows that this hypothesis
does not always hold. It is then shown that the theorem fails without the
hypothesis.
