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Summable function

From Encyclopedia of Mathematics
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A function , where is a space with a non-negative measure, for which the Lebesgue integral is defined and finite. The set of summable functions 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 . If , then is closed in the sense of uniform convergence.


Comments

A standard notation for the space of summable or Lebesgue integrable functions on is or (or , ).

References

[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 24
[a2] G.E. Shilov, B.L. Gurevich, "Integral, measure, and derivative: a unified approach" , Dover, reprint (1977) pp. 29ff (Translated from Russian)
How to Cite This Entry:
Summable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Summable_function&oldid=29893
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article