| Title:
|
An abstract and generalized approach to the Vitali theorem on nonmeasurable sets (English) |
| Author:
|
Basu, Sanjib |
| Author:
|
Sen, Debasish |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
65-70 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Here we present abstract formulations of two theorems of Solecki which deal with some generalizations of the classical Vitali theorem on nonmeasurable sets in spaces with transformation groups. (English) |
| Keyword:
|
spaces with transformation groups |
| Keyword:
|
$k$-additive measurable structure |
| Keyword:
|
$k$-small system |
| Keyword:
|
upper semicontinuous $k$-small system |
| Keyword:
|
$k$-additive algebra admissible with respect to a $k$-small system |
| MSC:
|
28A05 |
| MSC:
|
28D05 |
| idZBL:
|
07217180 |
| idMR:
|
MR4088693 |
| DOI:
|
10.21136/MB.2019.0116-17 |
| . |
| Date available:
|
2020-03-12T08:20:05Z |
| Last updated:
|
2020-11-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148064 |
| . |
| Reference:
|
[1] Erdős, P., Mauldin, R. D.: The nonexistence of certain invariant measures.Proc. Am. Math. Soc. 59 (1976), 321-322. Zbl 0361.28013, MR 0412390, 10.2307/2041493 |
| Reference:
|
[2] Johnson, R. A., Niewiarowski, J., Świątkowski, T.: Small systems convergence and metrizability.Proc. Am. Math. Soc. 103 (1988), 105-112. Zbl 0651.28004, MR 0938652, 10.2307/2047535 |
| Reference:
|
[3] Kharazishvili, A. B.: On some types of invariant measures.Sov. Math., Dokl. 16 (1975), 681-684 English. Russian original Translated from Dokl. Akad. Nauk SSSR 222 1975 538-540. Zbl 0328.28011, MR 0382603 |
| Reference:
|
[4] Kharazishvili, A. B.: Transformations Groups and Invariant Measures.Set-Theoretic Aspects. World Scientific, Singapore (1998). Zbl 1013.28012, MR 1692839, 10.1142/3810 |
| Reference:
|
[5] Niewiarowski, J.: Convergence of sequences of real functions with respect to small systems.Math. Slovaca 38 (1988), 333-340. Zbl 0659.28004, MR 0978763 |
| Reference:
|
[6] Pelc, A.: Invariant measures and ideals on discrete groups.Diss. Math. 255 (1986), 47 pages. Zbl 0625.04009, MR 0872392 |
| Reference:
|
[7] Riečan, B.: Abstract formulation of some theorems of measure theory.Mat.-Fyz. Čas., Slovensk. Akad. Vied 16 (1966), 268-273. Zbl 0174.34402, MR 0222235 |
| Reference:
|
[8] Riečan, B.: Abstract formulation of some theorems of measure theory. II.Mat. Čas., Slovensk. Akad. Vied 19 (1969), 138-144. Zbl 0193.00903, MR 0302848 |
| Reference:
|
[9] Riečan, B.: A note on measurable sets.Mat. Čas., Slovensk. Akad. Vied 21 (1971), 264-268. Zbl 0237.28001, MR 0306429 |
| Reference:
|
[10] Riečan, B., Neubrunn, T.: Integral, Measure and Ordering.Mathematics and Its Applications 411. Kluwer Academic Publisher, Dordrecht (1997). Zbl 0916.28001, MR 1489521, 10.1007/978-94-015-8919-2 |
| Reference:
|
[11] Riečanová, Z.: On an abstract formulation of regularity.Mat. Čas., Slovensk. Akad. Vied 21 (1971), 117-123. Zbl 0223.28001, MR 0304597 |
| Reference:
|
[12] Solecki, S.: Measurability properties of sets of Vitali's type.Proc. Am. Math. Soc. 119 (1993), 897-902. Zbl 0795.28010, MR 1152992, 10.2307/2160530 |
| Reference:
|
[13] Solecki, S.: On sets nonmeasurable with respect to invariant measures.Proc. Am. Math. Soc. 119 (1993), 115-124. Zbl 0784.28006, MR 1159177, 10.2307/2159832 |
| Reference:
|
[14] Vitali, G.: Sul problema della misura dei gruppi di punti di una retta. Nota.Gamberini e Parmeggiani, Bologna (1905), Italian \99999JFM99999 36.0586.03. |
| . |
