A. Pelc, Measure theory from the point of view of set theory, Wiadomosci Matematyczne 26 (1984), pp. 33-46 (in Polish).
We give a survey of some problems in measure theory, topology and set theory, chosen and studied from the point of view of foundations of mathematics. One group of such problems are those which belong to measure theory and topology but are independent of axioms of set theory. The other group contains problems in set theory, inspired by questions originating in measure theory. A classical example of the latter is the so called general measure problem.
A. Pelc, Idempotent ideals on abelian groups, Journal of Symbolic Logic 49 (1984), pp. 813-817.
An ideal I defined on a group G is called idempotent if for every element A of I, the set of g for which Ag-1 does not belong to I, is an element of I. We show that a countably complete idempotent ideal on an abelian group cannot be prime but may have strong saturation properties.
A. Pelc, The nonexistence of maximal invariant measures on abelian groups, Proceedings of the American Mathematical Society 92 (1984), pp. 55-57.
It is shown that every sigma-additive sigma-finite invariant measure on an abelian group has a proper sigma-additive invariant extension.
A. Pelc, Combinatorics on sigma-algebras and a problem of Banach, Fundamenta Mathematicae 123 (1984), pp. 1-9.
A sigma-algebra S on the reals is called measurable if there exists a probability measure on S vanishing on atoms of S. Banach asked if the union of two countably generated measurable sigma-algebras can generate a non-measurable sigma-algebra. This problem was solved positively by Grzegorek. Under the assumption of Martin's axiom we show a large family of sigma-algebras with the property that all its small subfamilies generate measurable sigma-algebras and all large subfamilies generate non-measurable sigma-algebras. We also consider a group-invariant version of Banach's problem and various questions concerning the structure of sigma-algebras and their measurability.