29 | 11 | 2023
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# Thomson's variational measure

## Abstract

For a local system$\small{\mathcal S}$ and a function f:R → R, Thomson defines (see B. S. Thomson, Real functions, Lect. Notes in Math., 1170, Springer-Verlag, 1985) the metric outer measure$\small{\mathcal S}\text{-}\small\mu _f$ on R.

We introduced in Real Functions - Current Topics the properties$\small{\mathcal S}AC ~({\mathcal S}$-absolute continuity) and $\small{\mathcal S}ACG$.

In this paper we show that for some particular local systems, such as $\small {\mathcal S}_o, ~{\mathcal S}_{ap}~ \text{and}~ {\mathcal S}_{\alpha ,\beta }\:$, there is a strong relationship between $\small{\mathcal S}\text{-}\mu _f~ \text{and} ~{\mathcal S}ACG$.

For example (see Theorem 8.4):

A function f:[a,b] → R is$\small{\mathcal S}_oACG$ on a Lebesgue measurable set E \subset [a,b]

$\small{\mathcal S}_o\text{-}\mu_f$ is absolutely continuous on E

⇒ f is VB*G ∩ (N) on E and f is continuous at each point of E

In Theorem 5.1, we show that:

A function f:[a,b]→ R, Lebesgue measurable on E ⊆ [a,b], is$\small {\mathcal S}_{ap}ACG$ on E if and only if$\small {\mathcal S}_{ap}\text{-}\mu _f$ is absolutely continuous on E.

This result remains true if$\small{\mathcal S}_{ap}$ is replaced by$\small{\mathcal S}_{\alpha ,\beta }$ with$\small\alpha ,\beta \in \bigl(1/2\,,\,1\bigr]$  (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).

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