Difference between revisions of "Luzin-C-property"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.N. Luzin, ''Mat. Sb.'' , '''28''' (1912) pp. 266–294</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.N. Luzin, "Collected works" , '''1''' , Moscow (1953) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Kamke, "Das Lebesgue–Stieltjes Integral" , Teubner (1960)</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.N. Luzin, ''Mat. Sb.'' , '''28''' (1912) pp. 266–294 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.N. Luzin, "Collected works" , '''1''' , Moscow (1953) (In Russian) {{MR|0059845}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}} {{ZBL|63.0183.05}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Kamke, "Das Lebesgue–Stieltjes Integral" , Teubner (1960) {{MR|0125193}} {{ZBL|0071.05401}} </TD></TR></table> |
Revision as of 12:12, 27 September 2012
A characteristic property of a measurable function that is finite almost-everywhere on its domain of definition. A function 
, finite almost-everywhere on 
, has the 
-property on 
if for every 
there is a perfect set 
in 
with measure 
on which 
is continuous if considered only on 
. The 
-property was introduced by N.N. Luzin [1], who also proved that for a function to have the 
-property it is necessary and sufficient that it be measurable and finite almost-everywhere on 
. This theorem of Luzin (the Luzin criterion) can be generalized to the case of functions of several variables (see [3], [4]) and is one of the main theorems in the metric theory of functions.
References
| [1] | N.N. Luzin, Mat. Sb. , 28 (1912) pp. 266–294 |
| [2] | N.N. Luzin, "Collected works" , 1 , Moscow (1953) (In Russian) MR0059845 |
| [3] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05 |
| [4] | E. Kamke, "Das Lebesgue–Stieltjes Integral" , Teubner (1960) MR0125193 Zbl 0071.05401 |
Comments
See Luzin criterion for additional references and comments.
How to Cite This Entry:
Luzin-C-property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin-C-property&oldid=28239
Luzin-C-property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin-C-property&oldid=28239
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article