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Article 41

Thomson's variational measure and some classical theorems

Real Analysis Exchange 25 (1999/2000), no. 2, 521-546.

Mathematical reviews subject classification: 26A45, 26A46, 26A24.

 

Abstract

Using the conditions increasing* and decreasing*, and Thomson's variational measure, we give an easy proof of the Denjoy-Lusin-Saks Theorem (see S. Saks, Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937, p. 230).

In Theorem 5.1 we extend (the function is not supposed to be continuous) Thomson's Theorems 44.1 and 44.2 (see B. S. Thomson, Real functions, Lect. Notes in Math., vol. 1170, Springer-Verlag, 1985), that are in a close relationship with the Denjoy-Lusin-Saks Theorem.

From this extension we obtain another classical result: the Denjoy-Young-Saks Theorem (see C. A. Faure, Sur le theoreme de Denjoy-Young-Saks, C. R. Acad. Sci. Paris  320 (1995), Serie I, 415-418).

As consequences of the Denjoy-Lusin-Saks Theorem we obtain two well-known results due to de la Vallee Poussin (see S. Saks, Theory of the integral, 2nd. rev. ed., vol. PWN,  Monografie Matematyczne, Warsaw, 1937, p. 125, 127).

 

We extend then these results (the set E used there is not only Borel, but also Lebesgue measurable) and give in Theorem 8.1 a de la Vallee Poussin type theorem for VB*G functions, that is in fact an extension of a result of Thomson (see B. S. Thomson, Real functions, Lect. Notes in Math., vol. 1170, Springer-Verlag, 1985, Theorem 46.3).

Finally, as a consequence of the previous results, we give a characterization for measurable functions that are VB*G ∩ N∞  on a Lebesgue measurable set.