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absolutely continuous axiom B₁ Baire set Banach space Boolean Borel equivalent Borel sets Borel subset bounded linear functional called Cauchy sequence closed sets Co(X compact Hausdorff space compact space continuous function continuous mapping continuous real-valued functions convex set Corollary countably compact Daniell integral E₁ E₂ elements finite measure finite number following proposition function f ƒ and g given Hausdorff space Hence homeomorphism integrable with respect isomorphism L₁ Lebesgue measure Lemma Let f Let ƒ linear manifold locally compact measurable function measurable sets measure algebra measure space measure zero metric space nonempty nonnegative functions nonnegative measurable function o-algebra o-finite one-to-one open intervals open set outer measure point mapping point of closure positive linear functional Problem Proof Prove rational numbers real numbers set containing set of finite set of measure Show topological space unique vector lattice x₁