On nonmeasurable sets and unions

In mathematics, a nonmeasurable set is one for which the "volume" cannot be assigned. This "volume" can be understood differently depending on the structure in which we search sucha a set. In fact, from the very beginning, the notation of nonmeasurable set was a source of considerable controversy. CONTENTS Introduction I 1. Preliminaries 2. nonmeasurable sets II 3. Kuratowski partitions 4. On Kuratowski partitions in tree structures 5. On Kuratowski partitions in Ellentuck topology 6. Ideals associated with Kuratowski partitions 7. Kuratowski partitions in Baire spaces 8. Kuratowski partitions and game theory 9. Kuratowski partitions in complete metric spaces 10. An example of a metric space without Kuratowski partitions III 11. The generalization of Louveau-Simpson Theorem 12. On the equivalences of Gitik-Shelak Theorem 13. The generalization of Halpern-Lauchli Theorem IV 14. {Partitions and point-finite covers in Baire spaces 15. Nonmeasurable unions for point-finite families 16. On the existence of measurable selectors Bibliography Index