Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published April 1968 | public
Journal Article

Countable and net convergence

Marsden, J. E.

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.

Additional Information

© 1968. Reprinted from the American Mathematical Monthly.

Additional details

Created:
August 19, 2023
Modified:
October 20, 2023