Let G be a Polish locally compact group acting on a Polish space X" role="presentation">X with a G-invariant probability measure μ" role="presentation">μ. We factorize the integral with respect to... Show moreLet G be a Polish locally compact group acting on a Polish space X" role="presentation">X with a G-invariant probability measure μ" role="presentation">μ. We factorize the integral with respect to μ" role="presentation">μ in terms of the integrals with respect to the ergodic measures on X, and show that Lp(X,μ)" role="presentation">Lp(X,μ) (1≤p<∞" role="presentation">1≤p<∞) is G-equivariantly isometrically lattice isomorphic to an Lp" role="presentation">Lp-direct integral of the spaces Lp(X,λ)" role="presentation">Lp(X,λ), where λ" role="presentation">λ ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of Lp(X,μ)" role="presentation">Lp(X,μ) as an Lp" role="presentation">Lp-direct integral of order indecomposable representations. If (X′,μ′)" role="presentation">(X′,μ′) is a probability space, and, for some 1≤q<∞" role="presentation">1≤q<∞, G acts in a strongly continuous manner on Lq(X′,μ′)" role="presentation">Lq(X′,μ′) as isometric lattice automorphisms that leave the constants fixed, then G acts on Lp(X′,μ′)" role="presentation">Lp(X′,μ′) in a similar fashion for all 1≤p<∞" role="presentation">1≤p<∞. Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If (X′,μ′)" role="presentation">(X′,μ′) is separable, the representation of G on Lp(X′,μ′)" role="presentation">Lp(X′,μ′) can then be disintegrated into order indecomposable representations. The notions of Lp" role="presentation">Lp-direct integrals of Banach spaces and representations that are developed extend those in the literature. Show less
