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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sur les fonctions de deux variables dont les coupes sont des dérivées

Author: Zbigniew Grande
Journal: Proc. Amer. Math. Soc. 57 (1976), 69-74
MSC: Primary 26A54
MathSciNet review: 0419702
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Abstract: This paper concerns relationships between the measurability of a function $ f:{I^2} \to R$ (when $ I = \langle 0,1\rangle $ and $ R$ is the set of all real numbers) and its cross sections $ {f_{{x_0}}}(y) = f({x_0},y)$ and $ {f^{{y_0}}}(x) = f(x,{y_0})$. A function $ g:I \to R$ is said to have property $ ({\mathbf{K}})$ if for each measurable set $ A \subset I$ of positive measure the function $ g$ is ponctuellement-discontinue (i.e., the set of continuities is dense) on the closure of the set of all density points of $ A$. The main result is: If a function $ f:{I^2} \to R$ is bounded and each $ {f_x}$ has property $ ({\mathbf{K}})$ and each $ {f^y}$ is a derivative, then $ f$ is Lebesgue measurable.

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Article copyright: © Copyright 1976 American Mathematical Society

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