Vol. 61, No. 1, 1975

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A ratio limit theorem for a strongly subadditive set function in a locally compact amenable group

John Cronan Kieffer

Vol. 61 (1975), No. 1, 183–190

It is the purpose of this paper to prove that the following property holds: Given a locally compact, amenable, unimodular group G, if S is a strongly subadditive, nonpositive, right invariant set function defined on the class 𝒦 of relatively compact Borel subsets of G, and if {Aα} is a net in 𝒦 satisfying an appropriate growth condition, then

lim λ(Aα)−1S(A α)

exists independently of {Aα}, where λ is Haar measure on G.

Mathematical Subject Classification 2000
Primary: 43A05
Secondary: 60B15, 28A65
Received: 15 May 1975
Published: 1 November 1975
John Cronan Kieffer