Geodesic
On Almost Uniform Convergence of Families of Functions
Canadian mathematical bulletin, Tome 7 (1964) no. 1, pp. 45-48

Voir la notice de l'article provenant de la source Cambridge

DOI

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. 
Zakon, Elias. On Almost Uniform Convergence of Families of Functions. Canadian mathematical bulletin, Tome 7 (1964) no. 1, pp. 45-48. doi: 10.4153/CMB-1964-005-0
@article{10_4153_CMB_1964_005_0,
     author = {Zakon, Elias},
     title = {On {Almost} {Uniform} {Convergence} of {Families} of {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {45--48},
     year = {1964},
     volume = {7},
     number = {1},
     doi = {10.4153/CMB-1964-005-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1964-005-0/}
}
TY  - JOUR
AU  - Zakon, Elias
TI  - On Almost Uniform Convergence of Families of Functions
JO  - Canadian mathematical bulletin
PY  - 1964
SP  - 45
EP  - 48
VL  - 7
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1964-005-0/
DO  - 10.4153/CMB-1964-005-0
ID  - 10_4153_CMB_1964_005_0
ER  - 
%0 Journal Article
%A Zakon, Elias
%T On Almost Uniform Convergence of Families of Functions
%J Canadian mathematical bulletin
%D 1964
%P 45-48
%V 7
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1964-005-0/
%R 10.4153/CMB-1964-005-0
%F 10_4153_CMB_1964_005_0

Cité par Sources :