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Difference between revisions of "Measurable function"

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Originally, a measurable function was understood to be a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632001.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632002.png" /> with the property that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632003.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632004.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632005.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632006.png" /> is a (Lebesgue-) [[Measurable set|measurable set]]. A measurable function on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632007.png" /> can be made continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632008.png" /> by changing its values on a set of arbitrarily small measure; this is the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320010.png" />-property of measurable functions (N.N. Luzin, 1913, cf. also [[Luzin-C-property|Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320011.png" />-property]]).
 
Originally, a measurable function was understood to be a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632001.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632002.png" /> with the property that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632003.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632004.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632005.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632006.png" /> is a (Lebesgue-) [[Measurable set|measurable set]]. A measurable function on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632007.png" /> can be made continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632008.png" /> by changing its values on a set of arbitrarily small measure; this is the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320010.png" />-property of measurable functions (N.N. Luzin, 1913, cf. also [[Luzin-C-property|Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320011.png" />-property]]).
  

Revision as of 20:48, 14 March 2012

2020 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL]

Originally, a measurable function was understood to be a function of a real variable with the property that for every the set of points at which is a (Lebesgue-) measurable set. A measurable function on an interval can be made continuous on by changing its values on a set of arbitrarily small measure; this is the so-called -property of measurable functions (N.N. Luzin, 1913, cf. also Luzin -property).

A measurable function on a space is defined relative to a chosen system of measurable sets in . If is a -ring, then a real-valued function on is said to be a measurable function if

for every real number , where

This definition is equivalent to the following: A real-valued function is measurable if

for every Borel set . When is a -algebra, a function is measurable if (or ) is measurable. The class of measurable functions is closed under the arithmetical and lattice operations; that is, if , are measurable, then , , , and ( real) are measurable; and are also measurable. A complex-valued function is measurable if its real and imaginary parts are measurable. A generalization of the concept of a measurable function is that of a measurable mapping from one measurable space to another.

References

[1] P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[2] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[3] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
How to Cite This Entry:
Measurable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_function&oldid=21670
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article