Generalized Haar integral

The 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) ∫ d S : C ( Ω , R ) → A ( S ) such that ∫ f d S = f for each f ∈ A ( S ) and for any f ∈ C ( Ω , R ) and s ∈ S , ∫ f s d S = ∫ f d S , where f s : Ω ∋ x ↦ f ( s x ) ∈ R .