Kieffer has considered the
problem of averaging strongly subadditive, nonpositive, right invariant set
functions S defined on the class 𝒦 of precompact Borel subsets of a locally
compact (unimodular) amenable group G as a means of defining entropy in a
abstract probabilistic context. He shows if {Aα} is a net in 𝒦 satisfying an
appropriate growth condition then λ(Aα)−1S(Aα) has a limit depending
only on S, where λ is right Haar measure on G. Here we prove a somewhat
stronger result of the same type based on a Fundamental Inequality valid
in any locally compact group G and for any set function S as described
above:
for all sets A in 𝒦 of positive measure and all open sets K in 𝒦 which satisfy
λ(K) = λ(K), the so-called open continuity sets in 𝒦.
