Vol. 15, No. 4, 1965

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Almost invariant measures

Ronald John Larsen

Vol. 15 (1965), No. 4, 1295–1305

Let μ be a regular complex-valued Borel measure on a locally compact topological (LC) group G which is finite on compact sets; and for each s G define the measure Tsμ by Tsμ(E) = μ(E + s), E Bc(G) the collection of all Borel subsets of G with compact closure. If f is a function on G then for each s G we set Tsf(t) = f(t + s), t G. Let X be a translation invariant subspace of C0(G), the space of continuous complex-valued functions on G which vanish at infinity, i.e., a subspace such that f X implies Tsf X, s G; and let U be an open symmetric neighborhood of zero in G. Then we shall say μ acts U-almost invariantly on X if G|h(t)|d|μ|(t) < , h X, and

∫              n∑      ∫
Gh(t)dTsμ(t) =    αi(s) G h(t)dTsiμ(t) (s ∈ U,h ∈ X),

where s1,s2,,sn are fixed elements of U. We shall say μ is a U-almost invariant measure on G if {Tsμs U} spans a finite dimensional space of measures. When U = G we shall say μ acts almost invariantly and μ is an almost invariant measure, respectively. The main results of this paper show that if μ acts U-almost invariantly on X then there exists some continuous function f such that

∫            ∫
Gh(t)dμ(t) = Gh (t)f(t)dm (t),  h ∈ X,

where dm is right invariant Haar measure on G; and that μ is a U-almost invariant measure if and only if there exists a continuous f such that (t) = f(t)dm(t) and {Tsfs U} spans a finite dimensional space of functions.

Mathematical Subject Classification
Primary: 28.75
Received: 18 December 1963
Revised: 6 June 1964
Published: 1 December 1965
Ronald John Larsen