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Separability and Scorza-Dragoni’s property. (English) Zbl 0581.28004
Let T be a compact Hausdorff topological space, \(\mu\) a Radon measure on T, X and Y two metric spaces. The main result of the paper states that if \(f: T\times X\to Y\) is a Carathéodory function, i.e. \(t\to f(t,x)\) is a \(\mu\)-measurable function for each \(x\in X\) and \(x\to f(t,x)\) is a continuous function for \(\mu\)-a.e. \(t\in T\), then f satisfies Scorza- Dragoni’s property, i.e. for every \(\epsilon >0\) there exists a closed subset \(T_{\epsilon}\) of T, with \(\mu (T\setminus T_{\epsilon})<\epsilon,\) such that the restriction of f to \(T_{\epsilon}\times X\) is a continuous function, provided that X is separable. Under the continuum hypothesis, it is proved that if X is not separable, then there exists \(f: [0,1]\times X\to {\mathbb{R}},\) a Carathéodory function with respect to Lebesgue measure, which does not satisfy Scorza-Dragoni’s property.

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A35 Measures and integrals in product spaces
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures