Vol. 61, No. 2, 1975

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Averaging strongly subadditive set functions in unimodular amenable groups. I

William Richard Emerson

Vol. 61 (1975), No. 2, 391–400

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:

λ(A )−1S(KA ) ≦ λ(K −1)−1lS(K )

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.

Mathematical Subject Classification 2000
Primary: 43A07
Received: 29 July 1975
Published: 1 December 1975
William Richard Emerson