# Thomson's variational measure

### Real Analysis Exchange 24 (1998/9), No. 2, 523-566.

#### Mathematical reviews subject classification: 26A45, 26A46, 26A24.

## Abstract

For a local system 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 on R.

We introduced in Real Functions - Current Topics the properties-absolute continuity) and .

In this paper we show that for some particular local systems, such as , there is a strong relationship between .

For example (see Theorem 8.4):

A function f:[a,b] → R is on a Lebesgue measurable set E \subset [a,b]

⇒ 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 on E if and only if is absolutely continuous on E.

This result remains true if is replaced by with (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).