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A lemma on set functions and its applications. (English) Zbl 0837.28011
Since the book “Matrix methods in analysis” (1985; Zbl 0564.46001) of the first author and C. Swartz, matrix methods are considered as a powerful tool in functional analysis and measure theory. They may be considered as an abstract formulation of the sliding hump argument. A variant of these methods was used by the reviewer [Rocky Mt. J. Math. 16, 253-275 (1986; Zbl 0604.28006)] replacing functions of the type $$A\to \varphi(A):= \sum_{i\in A} x_i$$ (for certain sets $$A$$ of natural numbers) by more general set functions $$\varphi(A)$$. In the paper under review, the authors establish a lemma about set functions which generalizes several matrix theorems and in particular reviewer’s version (in terms of set functions). The authors get as a simple consequence Rosenthal’s lemma. Several further applications are mentioned in the introduction.
Reviewer: H.Weber (Udine)

MSC:
 28B99 Set functions, measures and integrals with values in abstract spaces
Keywords:
matrix theorems; Rosenthal’s lemma