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Function decompositions related to the Luzin N-property. (Russian, English) Zbl 1054.26004
Sib. Mat. Zh. 45, No. 1, 178-188 (2004); translation in Sib. Math. J. 45, No. 1, 146-154 (2004).
The author considers the following question: What is the decomposition of a function into the sum of two functions such that
1) in the case of a monotone function it coincides with the Lebesgue decomposition into the sum of absolutely continuous and singular functions;
2) one of the summands possesses the N-property while the second one does not?
A class of completely regular functions is introduced, and it is shown that such functions possess the N-property. The author obtains a decomposition of an arbitrary continuous function into the sum of two functions, the first one of which is completely regular and the second one does not possess the N-property. A class is defined of strongly regular Borel functions for which the N-property is proven. From an arbitrary Borel function the author extracts a strongly regular function and a function that does not possess the N-property.
MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
60J55 Local time and additive functionals
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