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Scorza Dragoni type theorems. (English) Zbl 0744.28011
A Scorza-Dragoni type result verifies that if \(f(t,x)\) is a mapping on \(T\times X\), measurable in \(t\), and has a continuity property in \(x\), then the restriction of \(f\) to a set \(K\times K\) has the same continuity property, with \(K\subset T\) such that the measure of \(T\backslash K\) is arbitrarily small. Here \(T\) is a topological space with a measure on it, and \(X\) is a topological space. The paper establishes Scorza-Dragoni type results under quite relaxed topological conditions on \(T\), \(X\) and the range of \(f\), and for various continuity and semi-continuity properties on \(f\), which may also be a multifunction. Baire versions of the results are also provided.

MSC:
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
54C60 Set-valued maps in general topology
54C65 Selections in general topology
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