## 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
Full Text: