Thomsonʼs Variational Measure and Some Classical Theorems

Abstract

Using the conditions increasing$^*$ and decreasing$^*$, and Thomson's variational measure, we give an easy proof of the Denjoy-Lusin-Saks Theorem [12, 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 [13], that are closely related to the Denjoy-Lusin-Saks Theorem. From this extension we obtain another classical result: the Denjoy-Young-Saks Theorem [5]. As consequences of the Denjoy-Lusin-Saks Theorem we obtain two well-known results due to de la Vallée Poussin [12, p. 125, 127]. Then wee 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 $VB^*G$ functions, that is in fact an extension of a result of Thomson [13, Theorem 46.3]. Finally, we give characterizations for Lebesgue measurable functions that are $VB^*G \cap (N)$, and for measurable functions that are $VB^*G \cap N^{+\infty}$ on a Lebesgue measurable set.