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Thomson’s variational measure and some classical theorems. (English) Zbl 1010.26009

Summary: Using the conditions increasing\(^*\) and decreasing\(^*\), and Thomson’s variational measure, we give an easy proof of the Denjoy-Lusin-Saks Theorem [S. Saks, “Theory of the integral” (1937; Zbl 0017.30004), p. 230]. In Theorem 5.1 we extend (the function is not assumed to be continuous) Thomson’s Theorems 44.1 and 44.2 of [B. Thomson, “Real functions” (1985; Zbl 0581.26001)], that are closely related to the Denjoy-Lusin-Saks Theorem. From this extension we obtain another classical result: the Denjoy-Young-Saks Theorem [C.-A. Faure, C. R. Acad. Sci. Paris, Sér. I 320, No. 4, 415-418 (1995; Zbl 0835.26006)]. As consequences of the Denjoy-Lusin-Saks Theorem we obtain two well-known results due to de la Vallée Poussin [S. Saks, loc. cit., p. 125, 127]. Then we extend these results (the set \(E\) used there is not only Borel, but also Lebesgue measurable) and give in Theorem 8.1 a de la Vallée Poussin type theorem for \(\text{VB}^*\text{G}\) functions, that is in fact an extension of a result of B. S. Thomson [loc. cit., Theorem 46.3]. Finally, we give characterizations for Lebesgue measurable functions that are \(\text{VB}^*\text{G}\cap(N)\), and for measurable functions that are \(\text{VB}^*\text{G}\cap N^{+\infty}\) on a Lebesgue measurable set.

MSC:

26A45 Functions of bounded variation, generalizations
26A39 Denjoy and Perron integrals, other special integrals
26A46 Absolutely continuous real functions in one variable
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