A Characterization of Almost Uniform Convergence

1967 ◽  
Vol 74 (10) ◽  
pp. 1230
Author(s):  
N. Y. Luther
Keyword(s):  
Uniform Convergence ◽  
Almost Uniform Convergence

scholarly journals Noncommutative strong maximals and almost uniform convergence in several directions – Addendum

2021 ◽  
Vol 9 ◽  
Author(s):  
José M. Conde-Alonso ◽  
Adrián M. González-Pérez ◽  
Javier Parcet
Keyword(s):  
Uniform Convergence ◽  
Almost Uniform Convergence

On Almost Uniform Convergence of Families of Functions

1964 ◽  
Vol 7 (1) ◽  
pp. 45-48 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Uniform Convergence ◽  
Present Note ◽  
Measure Zero ◽  
Slight Modification ◽  
Real Parameter ◽  
General Proof ◽  
Almost Uniform Convergence

In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.

scholarly journals A characterization of the uniform convergence points set of some convergent sequence of functions

2021 ◽  
Vol 71 (2) ◽  
pp. 423-428
Author(s):  
Olena Karlova
Keyword(s):  
Uniform Convergence ◽  
Convergent Sequence ◽  
Normal Space ◽  
Disjoint Sequence ◽  
Isolated Points ◽  
Perfectly Normal Space ◽  
Set Of Points ◽  
Sequence Of Functions ◽  
Perfectly Normal

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

scholarly journals Characterization of pseudocompactness by the topology of uniform convergence on function spaces

1978 ◽  
Vol 26 (2) ◽  
pp. 251-256 ◽  
Author(s):  
R. A. McCoy
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Metrizable Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Tychonoff Space ◽  
Necessary And Sufficient

AbstractIt is shown that a Tychonoff space X is pseudocompact if and only if for every metrizable space Y, all uniformities on Y induce the same topology on the space of continuous functions from X into Y. Also for certain pairs of spaces X and Y, a necessary and sufficient condition is established in order that all uniformities on Y induce the same topology on the space of continuous functions from X into Y.

scholarly journals A note on convergence in measure and selection principles

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 473-477
Author(s):  
Dragan Djurcic ◽  
Ljubisa Kocinac
Keyword(s):  
Uniform Convergence ◽  
Almost Everywhere Convergence ◽  
Convergence In Measure ◽  
Mean Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Selection Principles ◽  
Almost Uniform Convergence ◽  
Special Modes

It is proved that some classes of sequences of measurable functions satisfy certain selection principles related to special modes of convergence (convergence in measure, almost everywhere convergence, almost uniform convergence, mean convergence).

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