Browsing by Author "Miller, Janice E."

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    Representation theory, Borel cross-sections, and minimal measures
    Miller, Janice E. (Virginia Tech, 1993)
    Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:Cb(E) → L∞(μ) with Γ(f∘Π) = f for every f ∈ Cb(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L∞(μ) with Γ(f∘Π) = f for every f in Cb(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.