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Start Over Subject Birkhoff integral

1. Convergence theorems for the Birkhoff integral

Creator:
Balcerzak, Marek and Potyrała, Monika
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Birkhoff integral, convergence theorems, and vector valued functions
Language:
English
Description:
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.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On integration of vector functions with respect to vector measures

Creator:
Rodríguez, José
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Bartle $^*$-integral, Dobrakov integral, McShane integral, Birkhoff integral, and $S^*$-integral
Language:
English
Description:
We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name $S^{*}$-integral. Our main result states that $S^{*}$-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public