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1997/1998 Characterizations of \(\mathbf{VBG ąp (N)}\)
Vasile Ene
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Real Anal. Exchange 23(2): 611-630 (1997/1998).


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).


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Vasile Ene. "Characterizations of \(\mathbf{VBG ąp (N)}\)." Real Anal. Exchange 23 (2) 611 - 630, 1997/1998.


Published: 1997/1998
First available in Project Euclid: 14 May 2012

MathSciNet: MR1639992

Primary: 26A39 , 26A45 , 26A46

Keywords: {\((PAC)\)} , {\(ACG\)} , {\(VBG\)} , {Lusin's condition \((N)\)}

Rights: Copyright © 1999 Michigan State University Press

Vol.23 • No. 2 • 1997/1998
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