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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1443–1457, November, 2004.
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Banakh, T.O., Kutsak, S.M., Maslyuchenko, V.K. et al. Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter. Ukr Math J 56, 1721–1737 (2004). https://doi.org/10.1007/s11253-005-0147-1
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DOI: https://doi.org/10.1007/s11253-005-0147-1