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On the measurability of functions of two variables

Published online by Cambridge University Press:  24 October 2008

Zbigniew Grande
Zbigniew Grande
Affiliation:
Mathematical Institute, The University, Gdańsk, Poland
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Let f be a real-valued function of two real variables. It is well known(5) that the Lebesgue measurability or even upper semicontinuity of all the sections fx and fy does not imply the measurability of f. Lipiński proved (3) that there exists a non-measurable function f such that each section fx and fy is Baire one, has the Darboux property, and fails to be approximately continuous at no more than one point. Davies (1) noted that the continuum hypothesis implies the existence of a non-measurable function f, such that each fy is measurable and each fx approximately continuous. Ursell (6) proved that if each fy is measurable and each fx continuous (or monotonic), then f is measurable.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 2 , March 1975 , pp. 335 - 336
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Davies, R. O.Separate approximate continuity implies measurability. Proc. Cambridge Philos. Soc. 73 (1973), 461–465.CrossRefGoogle Scholar
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(3)Lipiéski, J. S.On measurability of functions of two variables. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 131–135.Google Scholar
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