Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter |
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Authors: | T O Banakh S M Kutsak V K Maslyuchenko O V Maslyuchenko |
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Institution: | (1) Lviv National University, Lviv, Ukraine;(2) Chernivtsi National University, Chernivtsi, Ukraine |
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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.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1443–1457, November, 2004. |
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