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| − | ''of a locally summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579001.png" /> defined on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579002.png" />''
| + | #REDIRECT[[Lebesgue point]] |
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| − | The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579003.png" /> at which
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579004.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579005.png" /> is a closed cube containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579007.png" /> is the [[Lebesgue measure|Lebesgue measure]]. Here the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579008.png" /> can be real- or vector-valued.
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| − | ====Comments====
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| − | When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057900/l0579009.png" /> is real-valued and locally integrable, the complement of its Lebesgue set has (Lebesgue) measure zero. This is used in the study of differentiability via maximal functions, cf. [[#References|[a1]]].
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24</TD></TR></table>
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Latest revision as of 12:49, 7 August 2012
How to Cite This Entry:
Lebesgue set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_set&oldid=15208
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article