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- Title
Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter.
- Authors
Banakh, T.; Kutsak, S.; Maslyuchenko, V.; Maslyuchenko, O.
- 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.
- Subjects
BAIRE classes; REAL variables; INTEGRALS; INTEGRAL calculus; TOPOLOGICAL spaces; MATHEMATICAL functions
- Publication
Ukrainian Mathematical Journal, 2004, Vol 56, Issue 11, p1721
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-005-0147-1