Real Analysis Exchange

On the Measurability of Functions f:ℝ²→ℝ Having Pawlak’s Property in One Variable

Zbigniew Grande

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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.

Article information

Source
Real Anal. Exchange, Volume 25, Number 2 (1999), 647-652.

Dates
First available in Project Euclid: 3 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1230995399

Mathematical Reviews number (MathSciNet)
MR1778517

Subjects
Primary: 26B05: Continuity and differentiation questions 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

Keywords
Baire$^{**}_1$ property quasicontinuity Darboux property measurability section density topology function of two variables

Citation

Grande, Zbigniew. On the Measurability of Functions f :ℝ²→ℝ Having Pawlak’s Property in One Variable. Real Anal. Exchange 25 (1999), no. 2, 647--652. https://projecteuclid.org/euclid.rae/1230995399


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References

  • A. M. Bruckner, Differentiation of real functions, Lectures Notes in Math. 659, Springer-Verlag, Berlin 1978.
  • R. O. Davies, Approximate continuity implies measurability, Math. Proc. Camb. Philos. Soc. 73 (1973), 461–465.
  • Z. Grande, On strong quasi-continuity of functions of two variables, Real Anal. Exch., 21 (1995–96), 236–243.
  • Z. Grande, Un théorème sur la mesurabilité des fonctions de deux variables, Acta Math. Hung. 41 (1983), 89–91.
  • 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.
  • 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.