Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter
Abstract
We study the problem of the Baire classification of integrals g (y) = (If)(y) = ∫ X f(x, y)dμ(x), where y is a parameter that belongs to a topological space Y and f are separately continuous functions or functions similar to them. For a given function g, we consider the inverse problem of constructing a function f such that g = If. In particular, for compact spaces X and Y and a finite Borel measure μ on X, we prove the following result: In order that there exist a separately continuous function f : X × Y → ℝ such that g = If, it is necessary and sufficient that all restrictions g| Y n of the function g: Y → ℝ be continuous for some closed covering { Y n : n ∈ ℕ} of the space Y.
English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 11, pp 1721-1737.
Citation Example: Banakh T. O., Kutsak S. M., Maslyuchenko O. V., Maslyuchenko V. K. Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter // Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1443-1457.
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