| Title:
|
Extension of measures: a categorical approach (English) |
| Author:
|
Frič, Roman |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
130 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
397-407 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present a categorical approach to the extension of probabilities, i.e. normed $\sigma $-additive measures. J. Novák showed that each bounded $\sigma $-additive measure on a ring of sets $\mathbb{A}$ is sequentially continuous and pointed out the topological aspects of the extension of such measures on $\mathbb{A}$ over the generated $\sigma $-ring $\sigma (\mathbb{A})$: it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space $X$ over its Čech-Stone compactification $\beta X$ (or as the extension of continuous functions on $X$ over its Hewitt realcompactification $\upsilon X$). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that $\sigma (\mathbb{A})$ is the sequential envelope of $\mathbb{A}$ with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category $\mathop {{\mathrm ID}}$ of $-posets of fuzzy sets (such $-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on $\mathbb{A}$ over $\sigma (\mathbb{A})$ is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems. (English) |
| Keyword:
|
extension of measure |
| Keyword:
|
categorical methods |
| Keyword:
|
sequential continuity |
| Keyword:
|
sequential envelope |
| Keyword:
|
field of subsets |
| Keyword:
|
D-poset of fuzzy sets |
| Keyword:
|
effect algebra |
| Keyword:
|
epireflection |
| MSC:
|
18B30 |
| MSC:
|
28A05 |
| MSC:
|
28A12 |
| MSC:
|
28E10 |
| MSC:
|
54A40 |
| MSC:
|
54B30 |
| MSC:
|
54C20 |
| MSC:
|
60B99 |
| idZBL:
|
Zbl 1107.54014 |
| idMR:
|
MR2182385 |
| DOI:
|
10.21136/MB.2005.134212 |
| . |
| Date available:
|
2009-09-24T22:22:48Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134212 |
| . |
| Reference:
|
[1] Adámek, J.: Theory of Mathematical Structures.Reidel, Dordrecht, 1983. MR 0735079 |
| Reference:
|
[2] Bugajski, S.: Statistical maps I. Basic properties.Math. Slovaca 51 (2001), 321–342. Zbl 1088.81021, MR 1842320 |
| Reference:
|
[3] Bugajski, S.: Statistical maps II. Operational random variables.Math. Slovaca 51 (2001), 343–361. Zbl 1088.81022, MR 1842321 |
| Reference:
|
[4] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures.Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava, 2000. MR 1861369 |
| Reference:
|
[5] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics.Found. Phys. 24 (1994), 1331–1352. MR 1304942, 10.1007/BF02283036 |
| Reference:
|
[6] Frič, R.: Remarks on sequential envelopes.Rend. Istit. Math. Univ. Trieste 20 (1988), 19–28. MR 1013095 |
| Reference:
|
[7] Frič, R.: A Stone type duality and its applications to probability.Topology Proc. 22 (1999), 125–137. MR 1718934 |
| Reference:
|
[8] Frič, R.: Boolean algebras: convergence and measure.Topology Appl. 111 (2001), 139–149. Zbl 0977.54004, MR 1806034, 10.1016/S0166-8641(99)00195-9 |
| Reference:
|
[9] Frič, R.: Convergence and duality.Appl. Categorical Structures 10 (2002), 257–266. Zbl 1015.06010, MR 1916158, 10.1023/A:1015292329804 |
| Reference:
|
[10] Frič, R.: Łukasiewicz tribes are absolutely sequentially closed bold algebras.Czechoslovak Math. J. 52 (2002), 861–874. Zbl 1016.28013, MR 1940065, 10.1023/B:CMAJ.0000027239.28381.31 |
| Reference:
|
[11] Frič, R.: Measures on $\mathop {{\mathrm MV}}$-algebras.Soft Comput. 7 (2002), 130–137. 10.1007/s00500-002-0194-6 |
| Reference:
|
[12] Frič, R.: Duality for generalized events.Math. Slovaca 54 (2004), 49–60. Zbl 1076.22004, MR 2074029 |
| Reference:
|
[13] Frič, R.: Coproducts of $\mathop {\text{D}}$-posets and their applications to probability.Internt. J. Theoret. Phys. 43 (2004), 1625–1632. MR 2108299, 10.1023/B:IJTP.0000048808.83945.08 |
| Reference:
|
[14] Frič, R.: Remarks on statistical maps.(to appear). |
| Reference:
|
[15] Frič, R., Jakubík, J.: Sequential convergences on Boolean algebras defined by systems of maximal filters.Czechoslovak Math. J. 51 (2001), 261–274. MR 1844309, 10.1023/A:1013738728926 |
| Reference:
|
[16] Frič, R., McKennon, K., Richardson, G. D.: Sequential convergence in C(X).In: Convergence Structures and Applications to Analysis. (Abh. Akad. Wiss. DDR, Abt. Math.-Natur.-Technik 1979, 4N), Akademie-Verlag, Berlin, 1980, pp. 56–65. MR 0614001 |
| Reference:
|
[17] Gudder, S.: Fuzzy probability theory.Demonstratio Math. 31 (1998), 235–254. Zbl 0984.60001, MR 1623780 |
| Reference:
|
[18] Jurečková, M.: The measure extension theorem for $\mathop {{\mathrm MV}}$-algebras.Tatra Mountains Math. Publ. 6 (1995), 56–61. MR 1363983 |
| Reference:
|
[19] Kent, D. C., Richardson, G. D.: Two generalizations of Novák’s sequential envelope.Math. Nachr. 19 (1979), 77–85. MR 0563600, 10.1002/mana.19790910106 |
| Reference:
|
[20] Kôpka, F., Chovanec, F.: D-posets.Math. Slovaca 44 (1994), 21–34. MR 1290269 |
| Reference:
|
[21] Mišík, L., Jr.: Sequential completeness and $\lbrace 0,1\rbrace $-sequential completeness are different.Czechoslovak Math. J. 34 (1984), 424–431. MR 0761425 |
| Reference:
|
[22] Novák, J.: Ueber die eindeutigen stetigen Erweiterungen stetiger Funktionen.Czechoslovak Math. J. 8 (1958), 344–355. MR 0100826 |
| Reference:
|
[23] Novák, J.: On convergence spaces and their sequential envelopes.Czechoslovak Math. J. 15 (1965), 74–100. MR 0175083 |
| Reference:
|
[24] Novák, J.: On sequential envelopes defined by means of certain classes of functions.Czechoslovak Math. J. 18 (1968), 450–456. MR 0232335 |
| Reference:
|
[25 Papčo, M.] : On measurable spaces and measurable maps.Tatra Mountains Mathematical Publ. 28 (2004), 125–140. Zbl 1112.06005, MR 2086282 |
| Reference:
|
[26] Papčo, M.: On fuzzy random variables: examples and generalizations.(to appear). MR 2190258 |
| Reference:
|
[27] Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics.Kluwer Acad. Publ., Dordrecht, 1991. MR 1176314 |
| Reference:
|
[28] Riečan, B., Mundici, D.: Probability on $\mathop {{\mathrm MV}}$-algebras.In: Handbook of Measure Theory, Vol. II (Editor: E. Pap), North-Holland, Amsterdam, 2002, pp. 869–910. MR 1954631 |
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