Compactness and convergence of measures. (English) Zbl 0765.28008
The biting lemma of J. K. Brooks and R. V. Chacon, which is related to H. P. Rosenthal’s subsequence splitting lemma, is extended to Banach-valued measures. The \(\omega^ 2\)-convergence introduced by J. K. Brooks and R. V. Chacon [Adv. Math. 37, 16-26 (1980; Zbl 0463.28003)] is shown not to be topological. A sequence \((f_ n)\) in \(L^ 1(P)\) converges \(\omega^ 2\) to zero if and only if there exists a strictly positive function \(g\in L^ \infty(P)\) such that \((gf_ n)\) is weakly convergent to zero. A characterization of compact subsets of \(L^ 1(X)\) is given. Finally, the authors present an application of \(\omega^ 2\)- convergence to Schmeidler’s \(n\)-dimensional Fatou lemma.
Reviewer: H.-A.Klei (Paris)
MSC:
28B05 | Vector-valued set functions, measures and integrals |
28A33 | Spaces of measures, convergence of measures |
28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |