Real Analysis Exchange

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

Zbigniew Grande

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

First available in Project Euclid: 3 January 2009

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Mathematical Reviews number (MathSciNet)

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}

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


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.

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