Difference between revisions of "Luzin-C-property"
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| − | A characteristic property of a [[Measurable function|measurable function]] that is finite almost-everywhere on its domain of definition. A function | + | A characteristic property of a [[Measurable function|measurable function]] that is finite almost-everywhere on its domain of definition. A function $f$, finite almost-everywhere on $[0,1]$, has the $\mathcal{C}$-property on $[0,1]$ if for every $\epsilon>0$ there is a [[Perfect set|perfect set]] $Q$ in $[0,1]$ with measure $>1-\epsilon$ on which $f$ is continuous if considered only on $Q$. The $\mathcal{C}$-property was introduced by N.N. Luzin [[#References|[1]]], who also proved that for a function to have the $\mathcal{C}$-property it is necessary and sufficient that it be measurable and finite almost-everywhere on $[0,1]$. This theorem of Luzin (the [[Luzin criterion|Luzin criterion]]) can be generalized to the case of functions of several variables (see [[#References|[3]]], [[#References|[4]]]) and is one of the main theorems in the metric theory of functions. |
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Revision as of 06:43, 24 November 2012
A characteristic property of a measurable function that is finite almost-everywhere on its domain of definition. A function $f$, finite almost-everywhere on $[0,1]$, has the $\mathcal{C}$-property on $[0,1]$ if for every $\epsilon>0$ there is a perfect set $Q$ in $[0,1]$ with measure $>1-\epsilon$ on which $f$ is continuous if considered only on $Q$. The $\mathcal{C}$-property was introduced by N.N. Luzin [1], who also proved that for a function to have the $\mathcal{C}$-property it is necessary and sufficient that it be measurable and finite almost-everywhere on $[0,1]$. 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