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Luzin measurability of Carathéodory type mappings. (English) Zbl 1120.54010

Generalizing a result of [B. Ricceri and A. Villani, Matematiche 37 (1985), 156–161 (1982; Zbl 0581.28004)] the author proves that each \(k_{\mathbb R}\)- and \(\aleph_0\)-space \(X\) has the Scorza-Dragoni property, which means that for every Radon measure space \((T,\mu)\) and any metrizable space \(Y\) a function \(f:T\times X\to Y\) is Lusin measurable provided it is Carathéodory in the sense that \(f\) is continuous for every fixed \(t\in T\) and \(\mu\)-measurable for every fixed \(x\in X\). Such a function \(f:X\to Y\) is called (separately) Lusin measurable if for every \(\varepsilon>0\) there is a compact subset \(K\subset T\) with measure \(\mu(T\setminus K)<\varepsilon\) such that the restriction \(f| K\times X\) is (separately) continuous. It follows that each separable metrizable space has the Scorza-Dragoni property.
On the other hand, the author exposes a list of spaces without that property. This list includes: metric spaces of density \(\geq 2^{\aleph_0}\), the Sorgenfrey line, the function space \(C_p([0,1])\), the reals with the density topology, the Alexandroff duplicate of \([0,1]\), the two arrows space, the one-point compactification of a discrete space of cardinality \(\geq 2^{\aleph_0}\), the Fréchet-Urysohn fan \(S_\omega\), the Arens’ space, the sequential Arens’ space, etc.
One of the principal results of the paper is Theorem 4.2 asserting that for a Radon measure space \((T,\mu)\), a topological space \(X\), and a metrizable space \(Y\) a separately Lusin measurable function \(f:T\times X\to Y\) is Lusin measurable provided \(X\) is strongly functionally generated by a countable collection \(\mathcal C\) of its bounded subspaces. The latter means that for any discontinuous function \(g:X\to\mathbb R\) the restriction \(g| C\) is discontinuous for some set \(C\in\mathcal C\). Also the author studies the properties of the class of countable spaces with a unique non-isolated point, having the Scorza-Dragoni property.

MSC:

54C08 Weak and generalized continuity
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Citations:

Zbl 0581.28004
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References:

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