Real Analysis Exchange

On Borel measurable functions that are VBG and (N)

Vasile Ene

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The Banach-Zarecki Theorem states that \(VB \cap (N) = AC\) for continuous functions on a closed set. Hence it is a linear space. In this article we show that \(VB \cap (N)\) is a linear space on any real Borel set. Hence \(VBG \cap (N)\) will also be a real linear space for Borel measurable functions defined on an interval. As a consequence of this result, we show that the \(AK_N\) integral of Gordon (\cite{G14}) is well defined. We also give answers to Gordon’s questions in \cite{G14}.

Article information

Real Anal. Exchange, Volume 22, Number 2 (1996), 688-695.

First available in Project Euclid: 22 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A39: Denjoy and Perron integrals, other special integrals

\((N)\) \(VB\) \(VBG\) \(AC\) \(ACG\) \([{\mathcal C}G]\) Borel sets \({\mathcal C}_{ap}\)


Ene, Vasile. On Borel measurable functions that are VBG and (N). Real Anal. Exchange 22 (1996), no. 2, 688--695.

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