Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-16T22:26:15.688Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

Extract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Davies, R. O.Separate approximate continuity implies measurability. Proc. Cambridge Philos. Soc. 73 (1973), 461–465.CrossRefGoogle Scholar
(2)Grande, Z.Sur la mesurabilité des fonctions de deux variables. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 813–816.Google Scholar
(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
(4)Saks, S.Theory of the integral (Warsaw, 1937).Google Scholar
(5)Sierpiński, W.Sur un probléme concernant les ensembles mesurables superficiellement. Fund. Math. 1 (1920), 112–115.CrossRefGoogle Scholar
(6)Ursell, H. D.Some methods of proving measurability. Fund. Math. 32 (1939), 311–330.CrossRefGoogle Scholar