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| − | A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911202.png" /> is a space with a non-negative measure, for which the [[Lebesgue integral|Lebesgue integral]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911203.png" /> is defined and finite. The set of summable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911204.png" /> forms a linear subspace of the space of measurable functions. Taking the absolute value of a function and the maximum and minimum of a finite system of functions does not lead outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911205.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911206.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911207.png" /> is closed in the sense of [[Uniform convergence|uniform convergence]].
| + | #REDIRECT [[Absolutely integrable function]] |
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| − | ====Comments====
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| − | A standard notation for the space of summable or Lebesgue integrable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911208.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s0911209.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s09112010.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s09112011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091120/s09112012.png" />).
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 24</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.E. Shilov, B.L. Gurevich, "Integral, measure, and derivative: a unified approach" , Dover, reprint (1977) pp. 29ff (Translated from Russian)</TD></TR></table>
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Latest revision as of 09:53, 6 July 2013
How to Cite This Entry:
Summable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Summable_function&oldid=18514
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article