On the Borel classes of set-valued maps of two variables (Q2024130)

The main results generalize Lebesgue's theorem saying that \(f:\mathbb R\times \mathbb R \to \mathbb R\) which is of Borel class \(\alpha\) in the first and continuous in the second variable is of Borel class \(\alpha\). One of such results says in particular that each set-valued mapping \(F:X\times Y\rightsquigarrow Z\) which is of upper Borel class \(\alpha\) in the first and continuous (i.e., both upper and lower semicontinuous) in the second variable is of lower Borel class \(\alpha + 1\) under the assumptions that \(X\) is a perfect space, \(Y\) is a separable metric space, and \(Z\) is a perfectly normal space. The dual statement which we get interchanging the words lower and upper also holds. We only need the additional assumption that the values of \(F\) are countably compact. A statement saying that \(F\) is of lower Borel class \(\alpha + 2\) if \(F\) is of lower Borel class \(\alpha\) in the first variable and \(F\) is continuous in the second variable is also proved under some additional assumptions on the mapping and spaces.