An Improvement of a Recent Result of Thomson

Abstract

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}).