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The weak integral by partitions of unity. (English) Zbl 1423.28035
Let \((S,\Sigma,\mathcal{T},\mu)\) be a \(\sigma\)-finite outer regular quasi Radon measure space and \(X\) a Banach space. L. Di Piazza and V. Marraffa [Stud. Math. 151, No. 2, 175–185 (2002; Zbl 1005.28009)] studied for functions \(f:S\rightarrow X\) the \(\mathcal{PU}\)-integral (\(\mathcal{PU}\) for partition of the unity), a modification of the generalized McShane integral. Hereby the \(\mathcal{PU}\)-integral is approximated (with respect to the norm topology of \(X\)) by sums of the type \(\sum(\int\psi_id\mu)f(t_i)\) where \(0\leq\psi_i\in L^1(S,\mathbb{R})\) form a partition of the unity, i.e. \(\sum \psi_i=1\). In the article under review the author introduces the weak \(\mathcal{PU}\)-integral (using the weak topology of \(X\)). In particular it is proved that \(f:S\rightarrow X\) is weakly \(\mathcal{PU}\)-integrable if it is weakly McShane integrable and Pettis integrable.
Reviewer: Hans Weber (Udine)
28B05 Vector-valued set functions, measures and integrals
26A39 Denjoy and Perron integrals, other special integrals
46G10 Vector-valued measures and integration
Full Text: DOI Euclid