# On an improvement of a recent result of Thomson

### Real Analysis Exchange 25 (1999/2000), no. 1, 429-436.

#### Mathematical reviews subject classification: 26A45, 26A39, 28A12.

## Abstract

In *σ-finite Borel measures on the real line *(Real Analysis Exchange **23** (1997/8), no. 1, 185-192), Brian S. Thomson proved the following result:

Let f be AC*G on an interval [a,b]. Then the total variation measure μ = μ_{f} associated with f has the following properties:

- μ is a σ-finite Borel measure on [a,b];
- μ is absolutely continuous with respect to Lebesgue measure;
- There is a sequence of closed sets F
_{n} whose union is all of [a,b] such that μ (F_{n}) < ∞ for each n;
- for every Borel set B ⊂ [a,b].

Conversely, if a measure μ satisfies conditions 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 ∩ (N) on a Lebesgue measurable subset P of [a,b] and continuous at each point of P;
- the converse is also true even for μ defined on the Lebesgue measurable subsets of P (see Theorem 2 and the two examples in Remark 1).