| Title:
|
Convergence and submeasures in Boolean algebras (English) |
| Author:
|
Jech, Thomas |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
503-511 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A Boolean algebra carries a strictly positive exhaustive submeasure if and only if it has a sequential topology that is uniformly Fréchet. (English) |
| Keyword:
|
Boolean algebra |
| Keyword:
|
exhaustive submeasure |
| Keyword:
|
sequential topology |
| Keyword:
|
uniformly Fréchet topology |
| MSC:
|
03G05 |
| MSC:
|
28A60 |
| idZBL:
|
Zbl 06997366 |
| idMR:
|
MR3914716 |
| DOI:
|
10.14712/1213-7243.2015.262 |
| . |
| Date available:
|
2018-12-28T15:13:53Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147554 |
| . |
| Reference:
|
[1] Balcar B., Główczyński W., Jech T.: The sequential topology on complete Boolean algebras.Fund. Math. 155 (1998), no. 1, 59–78. MR 1487988 |
| Reference:
|
[2] Balcar B., Franek F., Hruška J.: Exhaustive zero-convergence structures on Boolean algebras.Acta Univ. Carolin. Math. Phys. 40 (1999), no. 2, 27–41. MR 1751539 |
| Reference:
|
[3] Balcar B., Jech T., Pazák T.: Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann.Bull. London Math. Soc. 37 (2005), no. 6, 885–898. MR 2186722, 10.1112/S0024609305004807 |
| Reference:
|
[4] Balcar B., Jech T.: Weak distributivity, a problem of von Neumann and the mystery of measurability.Bull. Symbolic Logic 12 (2006), no. 2, 241–266. Zbl 1120.03028, MR 2223923, 10.2178/bsl/1146620061 |
| Reference:
|
[5] Balcar B., Jech T.: Contributions to the theory of weakly distributive complete Boolean algebras.Andrzej Mostowski and foundational studies, IOS, Amsterdam, 2008, pages 144–150. MR 2422684 |
| Reference:
|
[6] Fremlin D. H.: Measure Algebras.Handbook of Boolean Algebras, 3, North-Holland Publishing, Amsterdam, 1989. MR 0991611 |
| Reference:
|
[7] Gaifman H.: Concerning measures on Boolean algebras.Pacific J. Math. 14 (1964), 61–73. Zbl 0127.02306, MR 0161952, 10.2140/pjm.1964.14.61 |
| Reference:
|
[8] Horn A., Tarski A.: Measures in Boolean algebras.Trans. Amer. Math. Soc. 64 (1948), 467–497. Zbl 0035.03001, MR 0028922, 10.1090/S0002-9947-1948-0028922-8 |
| Reference:
|
[9] Jech T.: Non-provability of Souslin's hypothesis.Comment. Math. Univ. Carolinae 8 (1967), no. 2, 291–305. MR 0215729 |
| Reference:
|
[10] Jech T.: Algebraic characterizations of measure algebras.Proc. Amer. Math. Soc. 136 (2008), no. 4, 1285–1294. MR 2367102, 10.1090/S0002-9939-07-09184-8 |
| Reference:
|
[11] Jech T.: Measures on Boolean algebras.Fund. Math. 239 (2017), no. 2, 177–183. MR 3681586, 10.4064/fm352-1-2017 |
| Reference:
|
[12] Kalton N.: The Maharam Problem.Publ. Math. Univ. Pierre et Marie Curie, 94, Univ. Paris VI, Paris, 1989. MR 1107322 |
| Reference:
|
[13] Kalton N. J., Roberts J. W.: Uniformly exhaustive submeasures and nearly additive set functions.Trans. Amer. Math. Soc. 278 (1983), 803–816. MR 0701524, 10.1090/S0002-9947-1983-0701524-4 |
| Reference:
|
[14] Kantorovič L. V., Vulih B. Z., Pinsker A. G.: Functional Analysis in Partially Ordered Spaces.Gosudarstv. Izdat. Tehn.-Teor. Lit., Moskva, 1950 (Russian). MR 0038006 |
| Reference:
|
[15] Kelley J. L.: Measures on Boolean algebras.Pacific J. Math. 9 (1959), 1165–1177. MR 0108570, 10.2140/pjm.1959.9.1165 |
| Reference:
|
[16] Maharam D.: An algebraic characterization of measure algebras.Ann. of Math. (2) 48 (1947), 154–167. MR 0018718, 10.2307/1969222 |
| Reference:
|
[17] Mauldin D., ed.: The Scottish Book.Birkhäuser, Boston, 1981. MR 0666400 |
| Reference:
|
[18] Monk J. D., Bonnet R., eds.: Handbook of Boolean algebras,.North-Holland Publishing, Amsterdam, 1989. MR 0991601 |
| Reference:
|
[19] Solovay R. M., Tennenbaum S.: Iterated Cohen extensions and Souslin's problem.Ann. of Math. (2) 94 (1971), 201–245. MR 0294139, 10.2307/1970860 |
| Reference:
|
[20] Suslin M.: Problème 3.Fund. Math. 1 (1920), 223. 10.4064/fm-1-1-223-224 |
| Reference:
|
[21] Talagrand M.: A simple example of pathological submeasure.Math. Ann. 252 (1979/80), no. 2, 97–102. MR 0593624, 10.1007/BF01420116 |
| Reference:
|
[22] Talagrand M.: Maharam's problem.Ann. of Math. (2) 168 (2008), no. 3, 981–1009. Zbl 1185.28002, MR 2456888, 10.4007/annals.2008.168.981 |
| Reference:
|
[23] Tennenbaum S.: Souslin's problem.Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 60–63. MR 0224456, 10.1073/pnas.59.1.60 |
| Reference:
|
[24] Todorčević S.: A dichotomy for P-ideals of countable sets.Fund. Math. 166 (2000), no. 3, 251–267. MR 1809418 |
| Reference:
|
[25] Todorcevic S.: A problem of von Neumann and Maharam about algebras supporting continuous submeasures.Fund. Math. 183 (2004), no. 2, 169–183. Zbl 1071.28004, MR 2127965, 10.4064/fm183-2-7 |
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