| Title:
|
On nonmeasurable images (English) |
| Author:
|
Rałowski, Robert |
| Author:
|
Żeberski, Szymon |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
423-434 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(X,\mathbb I)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R(A^2)$ is completely $\mathbb I $-nonmeasurable, i.e, it is $\mathbb I$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations $(R_\alpha )_{\alpha <2^\omega }.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak. (English) |
| Keyword:
|
nonmeasurable set |
| Keyword:
|
Bernstein set |
| Keyword:
|
Polish ideal space |
| MSC:
|
03E35 |
| MSC:
|
03E75 |
| MSC:
|
28A99 |
| idZBL:
|
Zbl 1224.03028 |
| idMR:
|
MR2657959 |
| . |
| Date available:
|
2010-07-20T16:49:22Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140579 |
| . |
| Reference:
|
[1] Cichoń, J., Jasiński, A.: A note on algebraic sums of sets of reals.Real Anal. Exchange 28 (2003), 493-497. MR 2010332, 10.14321/realanalexch.28.2.0493 |
| Reference:
|
[2] Cichoń, J., Morayne, M., Rałowski, R., Ryll-Nardzewski, Cz., {.Z}eberski, Sz.: On nonmeasurable unions.Topology and its Applications 154 (2007), 884-893. MR 2294636 |
| Reference:
|
[3] Ciesielski, K., Fejzić, H., Freiling, C.: Measure zero sets with non-measurable sum.Real Anal. Exchange 27 (2001/02), 783-793. MR 1923168 |
| Reference:
|
[4] Kharazishvili, A.: Some remarks on additive properties of invariant $\sigma$-ideals on the real line.Real Anal. Exchange 21 (1995/96), 715-724. Zbl 0879.28026, MR 1407284 |
| Reference:
|
[5] Kysiak, M.: Nonmeasurable algebraic sums of sets of reals.Colloquium Mathematicum 102 (2005), 113-122. Zbl 1072.28002, MR 2150273, 10.4064/cm102-1-10 |
| Reference:
|
[6] Rałowski, R., {.Z}eberski, Sz.: Complete nonmeasurability in regular families.Houston Journal in Mathematics 34 (2008), 773-780. MR 2448381 |
| Reference:
|
[7] Sierpiński, W.: Sur la question de la measurabilite de la base de M. Hamel.Fundamenta Mathematicae 1 (1920), 105-111 \JFM 47.0180.03. 10.4064/fm-1-1-105-111 |
| Reference:
|
[8] {.Z}eberski, Sz.: On completely nonmeasurable unions.Mathematical Logic Quarterly 53 (2007), 38-42. Zbl 1109.03046, MR 2288888, 10.1002/malq.200610024 |
| . |
