On the Measurability of Functions f:ℝ²→ℝ Having Pawlak’s Property in One Variable
Zbigniew Grande
Source: Real Anal. Exchange Volume 25, Number 2
(1999), 647-652.
Abstract
In this article we present a condition on the sections $f^y$ of a function $f: \mathbb{R}^2 \to \mathbb{R}$ having Lebesgue measurable sections $f_x$ and quasicontinuous sections $f^y$ which implies the measurability of $f$. This condition is more general than the Baire$^{**}_1$ property introduced by R. Pawlak in [7]. Some examples of quasicontinuous functions satisfying this condition and discontinuous on the sets of positive measure are given.
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Keywords: Baire$^{**}_1$ property; quasicontinuity; Darboux property; measurability; section; density topology; function of two variables
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.rae/1230995399
Mathematical Reviews number (MathSciNet): MR1778517
References
A. M. Bruckner, Differentiation of real functions, Lectures Notes in Math. 659, Springer-Verlag, Berlin 1978.
Mathematical Reviews (MathSciNet): MR507448
R. O. Davies, Approximate continuity implies measurability, Math. Proc. Camb. Philos. Soc. 73 (1973), 461–465.
Mathematical Reviews (MathSciNet): MR325870
Digital Object Identifier: doi:10.1017/S0305004100077033
Z. Grande, On strong quasi-continuity of functions of two variables, Real Anal. Exch., 21 (1995–96), 236–243.
Mathematical Reviews (MathSciNet): MR1377532
Zentralblatt MATH: 0851.26005
Z. Grande, Un théorème sur la mesurabilité des fonctions de deux variables, Acta Math. Hung. 41 (1983), 89–91.
Mathematical Reviews (MathSciNet): MR704527
Digital Object Identifier: doi:10.1007/BF01994065
S. Kempisty, Sur les fonctions quasi-continues, Fund. Math. 19 (1932), 184–197.
T. Neubrunn, Quasi-continuity, Real Anal. Exchange 14 (1988-89), 259–306.
Mathematical Reviews (MathSciNet): MR995972
Zentralblatt MATH: 0679.26003
R. Pawlak, On some class of functions intermediate between the family of continuous functions and the class ${\cal B}^*_1$, Abstract of $15^{th}$ Summer School on Real Functions Theory, Liptovský Ján, Slovakia, September 6–11, 1998.
S. Saks, Theory of the integral, Warsaw 1937.
W. Sierpiński, Sur un problème concernant les ensembles mesurables superficiellement, Fund. Math. 1 (1920), 112–115.
F. D. Tall, The density topology, Pacific J. Math. 62 (1976), 275–284.
Mathematical Reviews (MathSciNet): MR419709
Project Euclid: euclid.pjm/1102867878
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