Article 26

On Borel measurable functions that are VBG and (N)

Real Analysis Exchange 22 (1996-1997), no. 2, 688-695.

Mathematical reviews subject classification: 26A24; 26A39.

Abstract

The Banach-Zarecki Theorem states that VB ∩ (N) = AC for continuous functions on a closed set, hence it is a linear space.

In this article we show that VB ∩ (N) is a linear space on any real Borel set, hence VBG ∩ (N) will also be a real linear space for Borel functions defined on an interval.
         
As a consequence of this result, we can define a new integral which contains both, the PD-integral of Sarkhel and De (see The proximally continuous integrals, J. Austral. Math. Soc. (Series A) 31 (1981), 26-45) and the AK_N - integral of Gordon (see Some comments  on an approximately continuous Khintchine integral, Real Analysis Exchange 20 (1994/5), no. 2, 831-841).

We also give answers to Gordon's questions of Some comments on an approximately continuous Khintchine integral (Real Analysis Exchange 20 (1994/5), no. 2, 831-841):

1. 1. Is every VBG ∩ (N) ∩ C_ap function a [CG] function?
   2. Is every indefinite AP integral a [CG] function?

The answer to Question 1 is negative and the answer to Question 2 is affirmative (see Remark 4).