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Measurability of functions with approximately continuous vertical sections and measurable horizontal sections. (English) Zbl 0852.28004

It is well-known that each function \(f: \mathbb{R}^2\to \mathbb{R}\) with continuous vertical sections and measurable horizontal sections is measurable as a function of two variables. The authors show that this fact is independent of set theory if we replace continuous by approximately continuous. Main results: 1. Let \(f: \mathbb{R}^2\to \mathbb{R}\) be a function with approximately continuous vertical sections and Baire 1 horizontal sections. Then \(f\) is Baire 2. 2. Suppose there exists a real-valued measurable cardinal. Then for every function \(f: \mathbb{R}^2\to \mathbb{R}\) and \(\alpha< \omega_1\), if all vertical sections of \(f\) are approximately continuous and all horizontal sections of \(f\) are Baire \(\alpha\), then \(f\) is Baire \(\alpha+ 1\). 3. Suppose that \(\mathbb{R}\) can be covered by \(\omega_1\) closed null sets. Then there exists a nonmeasurable function \(f: \mathbb{R}^2\to \mathbb{R}\) such that each section \(f_x\) is approximately continuous and each section \(f^y\) is Baire 2. 4. In the random real model, every function \(f: \mathbb{R}^2\to \mathbb{R}\) with approximately continuous vertical sections and measurable horizontal sections is measurable as a function of two variables.

MSC:

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
03E35 Consistency and independence results
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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