Geodesic
Convergence theorems for the Birkhoff integral
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1207-1219 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence $(f_n)$ of functions from a measure space to a Banach space. In one result the equi-integrability of $f_n$'s is involved and we assume $f_n\to f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $(f_n)$ to $f$ is assumed.
We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence $(f_n)$ of functions from a measure space to a Banach space. In one result the equi-integrability of $f_n$'s is involved and we assume $f_n\to f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $(f_n)$ to $f$ is assumed.
Classification : 28B05, 46G10
Keywords: Birkhoff integral; convergence theorems; vector valued functions 
@article{CMJ_2008_58_4_a24,
     author = {Balcerzak, Marek and Potyra{\l}a, Monika},
     title = {Convergence theorems for the {Birkhoff} integral},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1207--1219},
     year = {2008},
     volume = {58},
     number = {4},
     mrnumber = {2471177},
     zbl = {1174.28011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a24/}
}
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Balcerzak, Marek; Potyrała, Monika. Convergence theorems for the Birkhoff integral. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1207-1219. http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a24/