Generalized Haar integral

AbstractThe aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) ∫dS:C(Ω,R)→A(S) such that ∫fdS=f for each f∈A(S) and for any f∈C(Ω,R) and s∈S, ∫fsdS=∫fdS, where fs:Ω∋x↦f(sx)∈R