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The Vitali-Hahn-Saks theorems for k-triangular set functions. (English) Zbl 0626.28001
For an algebra $${\mathcal A}$$ of sets and $$k\in (0,\infty)$$, a set function $$\mu:{\mathcal A}\to {\mathbb{R}}_+$$ is said to be k-triangular if $$| \mu (A\cup B)-\mu (A)| \leq k\mu (B)$$ holds for each pair of disjoint sets $$A,B\in {\mathcal A}.$$ The author proves variants of the Vitali-Hahn-Saks (or Brooks-Jewett) Theorem for exhaustive k-triangular set functions on an algebra having the subsequential interpolation property and for k- triangular set functions of regular variation on the Borel subsets of a Hausdorff locally compact topological space. The paper contains numerous references on the subject; two preprints quoted by the author are now published: P. de Lucia and P. Morales [Ric. Mat. 35, 75-87 (1986; Zbl 0612.28006)] and H. Weber [Rocky Mt. J. Math. 16, 253- 275 (1986; Zbl 0604.28006)].
Reviewer: Klaus D.Schmidt

##### MSC:
 28A10 Real- or complex-valued set functions 28A33 Spaces of measures, convergence of measures 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28B10 Group- or semigroup-valued set functions, measures and integrals