On the measurability of functions with quasi-continuous and upper semi-continuous vertical sections

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• 作者：Zbigniew Grande (1)
  
• 关键词：Primary 26B05 ; 28A10 ; 28A05 ; Lebesgue measurability ; Baire property ; Baire classes ; upper semi ; continuity ; quasi ; continuity ; sup ; measurability
• 刊名：Mathematica Slovaca
• 出版年：2013
• 出版时间：August 2013
• 年：2013
• 卷：63
• 期：4
• 页码：793-798
• 全文大小：158KB
• 参考文献：1. BRUCKNER, A. M.: / Differentiation of Real Functions. Lecture Notes in Math. 659, Springer-Verlag, Berlin, 1978.
  2. DAVIES, R. O.: / Separate approximate continuity implies measurability, Proc. Cambridge Philos. Soc. 73 (1973), 461-65. CrossRef
  3. GRANDE, Z.: / Quelques remarques sur la semi-continuité supérieure, Fund. Math. 126 (1985), 1-3.
  4. KEMPISTY, S.: / Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184-97.
  5. LACZKOVICH, M.-MILLER, A.: / Measurability of functions with approximately continuous vertical sections and measurable horizontal sections, Colloq. Math. 69 (1995), 299-08.
  6. MARCZEWSKI, E.-RYLL-NARDZEWSKI, CZ.: / Sur la mesurabilité des fonctions de plusieurs variables, Ann. Polon. Math. 25 (1952), 145-49.
  7. NEUBRUNN, T.: / Quasi-continuity, Real Anal. Exchange 14 (1988/89), 259-06.
  8. SIERPI?SKI, W.: / Sur un problème concernant les ensembles mesurable superficiellement, Fund. Math. 1 (1920), 112-15.
  9. SIKORSKI, R.: / Real Functions, PWN, Warsaw, 1957 (Polish).
  10. SZRAGIN, W.: / The conditions for the measurability of the superposition Dokl. Akad. Nauk SSSR 197 (1971), 295-98 (Russian).
  
• 作者单位：Zbigniew Grande (1)
  
  1. Institute of Mathematics, Kazimierz Wielki University, Plac Weyssenhoffa 11, 85-072, Bydgoszcz, Poland
  

文摘

Let f: ?sup class="a-plus-plus">2 ??be a function with upper semicontinuous and quasi-continuous vertical sections f x (t) = f(x, t), t, x ?? It is proved that if the horizontal sections f y (t) = f(t, y), y, t ?? are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].