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Countable and net convergence

Marsden, J. E. (1968) Countable and net convergence. American Mathematical Monthly, 75 (4). ISSN 0002-9890. http://resolver.caltech.edu/CaltechAUTHORS:20100824-104216924

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Abstract

It is well known that Lebesgue's dominated convergence theorem does not hold for nets; that is, having a countable sequence is essential. On the other hand, for a real valued function on an interval, sequences do suffice; that is, lim_(x → y)f(x) = a iff lim_(n → ∞)f(x_n) = a for every sequence x_n → y. The purpose of this note is to isolate the basic reasons for these phenomena.


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Additional Information:© 1968. Reprinted from the American Mathematical Monthly.
Record Number:CaltechAUTHORS:20100824-104216924
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100824-104216924
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ID Code:19628
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 16:48
Last Modified:26 Dec 2012 12:21

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