Previous |  Up |  Next

Article

Title: Extension of measures: a categorical approach (English)
Author: Frič, Roman
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
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
.
Date available: 2009-09-24T22:22:48Z
Last updated: 2015-11-01
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
.

Files

Files Size Format View
MathBohem_130-2005-4_6.pdf 297.5Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo