Open Access
1999/2000 Thomsonʼs Variational Measure and Some Classical Theorems
Vasile Ene
Real Anal. Exchange 25(2): 521-546 (1999/2000).


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.


Download Citation

Vasile Ene. "Thomsonʼs Variational Measure and Some Classical Theorems." Real Anal. Exchange 25 (2) 521 - 546, 1999/2000.


Published: 1999/2000
First available in Project Euclid: 3 January 2009

zbMATH: 1010.26009
MathSciNet: MR1778509

Primary: 26A24 , 26A45 , 26A46

Keywords: $F$-null sets , $VB^*G$ , Lusin's condition $(N)$ , the condition increasing$^*$ , Thomson's variational measure

Rights: Copyright © 1999 Michigan State University Press

Vol.25 • No. 2 • 1999/2000
Back to Top