30 | 05 | 2024


About Vasile Ene

This email address is being protected from spambots. You need JavaScript enabled to view it.

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.



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 AKN - 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. Is every VBG ∩ (N) ∩ Cap 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).