A note on invariant integrals on locally compact semigroups
Loren N. Argabright · Proceedings of the American Mathematical Society · 1966
An integral on a locally compact (Hausdorff) semigroup 5 is a nontrivial, positive, linear functional I on the space CiS) of continuous real-valued functions on 5 with compact supports.If 5 satisfies the condition (#) for each compact set AC5 and each element aES, the set Aa-1 = {xES\xaEK} is compact; then, whenever fECiS) and aES, the function /" defined by/a(x) =fixa) is also in CiS).In this case, an integral / on 5 is called right invariant provided I(J) =I(Ja) for all fECiS) and all aES.A regular Borel measure p.on S is called r*-invariant if ix(Pa_1) =ju(P) for all Borel sets B and all aES.J. H. Michael [7] introduced the above concept of an invariant integral1 and proved that if 5 contains a unique minimal left ideal and satisfies some additional conditions (see [7]) then 5 admits a right invariant integral.P. S. Mostert [8] then pointed out that Michael's conditions could be weakened considerably and he also gave a much shorter proof.In this note we prove the following: