[For the entire collection see Zbl 0714.00017.]
The author announces some versions of two classical boundedness theorems due to Nikodým and Dieudonné. They involve special systems of sets, rather technical substitutes for additivity of set functions and nonstandard types of boundedness of families of set functions. Most of the respective notions are due to the author. For closely related material see also his book “Introduction to the theory of set functions” (Russian) (1989; Zbl 0732.28001), Chapter 3, and his recent paper in Mat. Sb. 183, No. 6, 155-176 (1992).
{Reviewer’s remarks: (1) Example 1.3 is well-known; cf. F. J. Freniche [Proc. Am. Math. Soc. 92, 362-366 (1984; Zbl 0529.28004)]. (2) Propositions 1.1 and 1.2 are taken from D. Candeloro [Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Math. 9, 249-260 (1985; Zbl 0587.28006).}.