## Real Analysis Exchange

### Characterizations of $\mathbf{VBG ąp (N)}$

Vasile Ene

#### Abstract

We show that $VBG \cap (N)$ is equivalent with Sarkhel and Kar’s class $(PAC)G$ on an arbitrary real set. Hence $VBG \cap (N)$ is an algebra on that set. In Theorem 4, we give three characterizations for $VBG \cap (N)$ on an arbitrary real set. It follows that Gordon’s $AK_N$-integral is a special case of the $PD$-integral of Sarkhel and De (Remark 3). In Theorem 3 we obtain the following surprising result: a Lebesgue measurable function $f$ is $VBG$ on $E$ if and only if $f$ is $VBG$ on any null subset of $E$. We also find seven characterizations of $VBG ąp (N)$ for Lebesgue measurable functions (see Theorem 5). For continuous functions on a closed set, we obtain several characterizations of the class $ACG$. Using a different technique, we obtain other characterizations of $VBG \cap (N)$ for a Lebesgue measurable function (see Theorem 8).

#### Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 611-630.

Dates
First available in Project Euclid: 14 May 2012

Ene, Vasile. Characterizations of $\mathbf{VBG ąp (N)}$. Real Anal. Exchange 23 (1999), no. 2, 611--630. https://projecteuclid.org/euclid.rae/1337001369