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1997/1998 Characterizations of \(\mathbf{VB^*G} \cap \mathbf{(N)}\)
Vasile Ene
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Real Anal. Exchange 23(2): 571-600 (1997/1998).


We introduce the condition \((PAC^*)\) that is a slight modification of the condition \((PAC)\) of Sarkhel and Kar. The main result is Theorem 4: \emph{A function \(f:[a,b] \to {\mathbb R}\) is \(VB^*G ąp (N)\) on a subset \(E\) of \([a,b]\) if and only if \(f \in (PAC^*)\) on \(E\)}. Consequently, \emph{the set \(\{f:[a,b] \to {\mathbb R}: f \in VB^*G ąp (N)\) on \(E\}\) is an algebra, whenever \(E\) is a subset of \([a,b]\)}. Using Theorem 1, we find seven characterizations of \(VB^*G ąp (N)\) on a Lebesgue measurable set (Theorem 5). We also give fifteen characterizations of the class of \(AC^*G\) functions on a closed set \(E\), that are continuous at each point of \(E\) (Theorem 6). In the last two sections, using Thomson’s outer measure \({\mathcal S}_o\text{-}\mu_f\), we characterize a \(VB^*G ąp (N)\) function \(f\) on a Lebesgue measurable set (Theorem 9). As a consequence we obtain that: \emph{A function \(f:[a,b] \to {\mathbb R}\) is \(AC^*G\) on a closed subset \(E\) of \([a,b]\) and continuous at each point of \(E\) if and only if \({\mathcal S}_o\text{-}\mu_f(Z) = 0\) whenever \(Z\) is a null subset of \(E\)} (Theorem 10).


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Vasile Ene. "Characterizations of \(\mathbf{VB^*G} \cap \mathbf{(N)}\)." Real Anal. Exchange 23 (2) 571 - 600, 1997/1998.


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

MathSciNet: MR1639984

Primary: 26A45 , 26A46

Keywords: {\((PAC)\)} , {\(AC^*G\)} , {\(VB^*G\)} , {Lusin's condition \((N)\)}

Rights: Copyright © 1999 Michigan State University Press

Vol.23 • No. 2 • 1997/1998
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