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On a correlation of A- and B-integrals. (Russian) Zbl 0606.26003

In 1919 A. Denjoy [C. R. Acad. Sci., Paris 169, 219-221 (1919)] gave the definition of B-integral, which is an extension of Lebesgue integral: let f be given almost everywhere on \((-\infty,+\infty)\) with a period 1. Denote \(S(t)=\sum^{n}_{k=1}f(t+x_{nk})(t_ k-t_{k- 1}),\) where \(0=t_ 0<t_ 1<...<t_ n=1,\quad t_{k-1}\leq x_{nk}<t_ k.\) We say that f is a B-integrable function on \([0,1)\) if there exists the limit (in measure on [0,1)) \(\lim_{\max (t_ k- t_{k-1})\to 0}S(t)=(B)\int^{1}_{0}f(x)dx\) uniformly over \(x_{nk}\). On the other hand, in 1928 E. C. Titchmarsh [Proc. Lond. Math. Soc. 29, 49-80 (1928)] considered the A-integral. A measurable function f on [0,1) is said to be A-integrable if \(mes\{x\in [0,1);\quad | f(x)| >n\}=o(1/n)\) and there exists \(\lim_{n\to \infty}\int_{0\leq x<1;\quad | f(x)| \leq n}f(x)x=(A)\int^{1}_{0}f(x)dx.\)
In the present paper for the class of measurable functions the fact that every B-integrable function is A-integrable is proved.
Reviewer: T.I.Akhobadze

MSC:

26A39 Denjoy and Perron integrals, other special integrals
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