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A note on Carathéodory type function. (English) Zbl 0809.28007

The following result is proved: Let \((T,{\mathcal A},\mu)\) be a measure space, \((X,{\mathcal B})\) a measurable space, and \(f: T\times X\to Y\) a function with \(Y\) a topological space. If \(f\) is measurable with respect to the product of the \(\mu\)-completion of \(\mathcal A\) and \(\mathcal B\), and if \(Y\) has a countable base, then a set \(S\in {\mathcal A}\) exists, with \(\mu(T\backslash S)= 0\) and \(f\) restricted to \(S\times X\) is measurable with respect to the product of \(\mathcal A\) and \(\mathcal B\). Several implications of this result in the direction of Scorza-Dragoni type results are indicated.

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
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