2015/07/09 by Robert Rałowski, Robert Ralowski, Ralowski, Robert +3
Computer Science · Mathematics · #03E50 #03E75 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Primary: 03E17 #Rings, Modules, and Algebras #Secondary: 28A99 #math.GN #msc:03E17 #msc:03E50 #msc:03E75 #msc:28A99
paper · pdf · doi:10.48550/arxiv.1507.02496
13 pages
arxiv created 2015/07/09 · openalex publication_date 2015/07/09 · arxiv updated 2015/07/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s0. We will consider also ideal l0 defined by Laver trees and m0 defined by Miller trees. With the mentioned ideals one can consider s, l and m-measurablility. We have shown that there exists a subset A of the Baire space which is s, l and m nonmeasurable at the same time. Moreover, A forms m.a.d. family which is also dominating. We show some examples of subsets of the Baire space which are measurable in one sense and nonmeasurable in the other meaning. We also examine terms nonmeasurable and completely nonmeasurable (with respect to several ideals with Borel base). There are several papers about finding (completely) nonmeasurable sets which are the union of some family of small sets. In this paper we want to focus on the following problem: "Let P be a family of small sets. Is it possible that for all A which is a subset of P, union of A is nonmeasurable implies that union of A is completely nonmeasurable?" We will consider situations when P is a partition of R, P is point-finite family and P is point-countable family. We give an equivalent statement to CH using terms nonmeasurable and completely nonmeasurable.