Thomsonʼs Variational Measure

Abstract

For a local system ${\mathcal S}$ and a function $f:{\mathbb R} \to {\mathbb R}$, Thomson defines in \cite{T5} the metric outer measure ${\mathcal S}\text{-}\mu_f$ on ${\mathbb R}$, and in \cite{Ene1}, we introduce the properties ${\mathcal S}AC$ (${\mathcal S}$-absolute continuity) and ${\mathcal S}ACG$. In this paper we show that for some particular local systems, such as ${\mathcal S}_o$, ${\mathcal S}_{ap}$ and ${\mathcal S}_{\alpha,\beta}$, there is a strong relationship between ${\mathcal S}\text{-}\mu_f$ and ${\mathcal S}ACG$. For example: \emph{a function $f:[a,b] \to {\mathbb R}$ is ${\mathcal S}_oACG$ on a Lebesgue measurable set $E \subset [a,b]$ $\Longleftrightarrow$ ${\mathcal S}_o\text{-}\mu_f$ is absolutely continuous on $E$ $\Longleftrightarrow$ $f$ is $VB^*G \cap (N)$ on $E$ and $f$ is continuous at each point of $E$} (see Theorem 8.4). In Theorem 5.1, we show that \emph{a function $f:[a,b] \to {\mathbb R}$, Lebesgue measurable on $E \subseteq [a,b]$, is ${\mathcal S}_{ap}ACG$ on $E$ if and only if ${\mathcal S}_{ap}\text{-}\mu_f$ is absolutely continuous on $E$}. This result remains true if ${\mathcal S}_{ap}$ is replaced by ${\mathcal S}_{\alpha,\beta}$ with $\alpha,\beta \in (1/2,1]$ (see Theorem 5.1). We also give several characterizations of the $VB^*G$ functions on an arbitrary set, that are continuous at each point of that set (see Theorem 7.4).