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1999/2000 An Improvement of a Recent Result of Thomson
Vasile Ene
Real Anal. Exchange 25(1): 429-436 (1999/2000).


In \cite{T13}, Brian S. Thomson proved the following result: \emph{Let $f$ be $AC^*G$ on an interval $[a,b]$. Then the total variation measure $\mu = \mu_f$ associated with $f$ has the following properties: a) $\mu$ is a $\sigma$-finite Borel measure on $[a,b]$; b) $\mu$ is absolutely continuous with respect to Lebesgue measure; \linebreak c) There is a sequence of closed sets $F_n$ whose union is all of $[a,b]$ such that $\mu(F_n) < \infty$ for each $n$; d) $\mu(B) = \mu_f(B) = \int_B|f^\prime(x)|\, dx$ for every Borel set $B \subset [a,b]$. Conversely, if a measure $\mu$ satisfies conditions \linebreak a)--c) then there exists an $AC^*G$ function $f$ for which the representation d) is valid.} In this paper we improve Thomson's theorem as follows: in the first part we ask $f$ to be only $VB^*G \cap (N)$ on a Lebesgue measurable subset $P$ of $[a,b]$ and continuous at each point of $P$; the converse is also true even for $\mu$ defined on the Lebesgue measurable subsets of $P$ (see Theorem \ref{T2} and the two examples in Remark~\ref{R1}).


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Vasile Ene. "An Improvement of a Recent Result of Thomson." Real Anal. Exchange 25 (1) 429 - 436, 1999/2000.


Published: 1999/2000
First available in Project Euclid: 5 January 2009

zbMATH: 1015.26020
MathSciNet: MR1758899

Primary: 26A39 , 26A45 , 28A12

Keywords: $AC^*G$ , $VB^*G$ , Borel sets , Lebesgue sets , Lusin's condition $(N)$ , variational measure

Rights: Copyright © 1999 Michigan State University Press

Vol.25 • No. 1 • 1999/2000
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