In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a σ-field and the measure on it. In general, the existence of the measure on σ-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a σ-field. Therefore, in this article we define a measure (pre-measure) on a semialgebra and extend it to a measure on a σ-field. Furthermore, we give a σ-measure as an extension of the measure on a σ-field. We follow [24], [10], and [31].
@article{bwmeta1.element.doi-10_1515_forma-2015-0025,
author = {Noboru Endou},
title = {Construction of Measure from Semialgebra of Sets1},
journal = {Formalized Mathematics},
volume = {23},
year = {2015},
pages = {309-323},
zbl = {1334.28002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2015-0025}
}
Noboru Endou. Construction of Measure from Semialgebra of Sets1. Formalized Mathematics, Tome 23 (2015) pp. 309-323. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2015-0025/
[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
[2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.
[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990. | Zbl 06213858
[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.
[6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.
[7] Józef Białas. Properties of Caratheodor’s measure. Formalized Mathematics, 3(1):67–70, 1992.
[8] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.
[9] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.
[10] V.I. Bogachev. Measure Theory, volume 1. Springer, 2006.
[11] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.
[12] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.
[13] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
[14] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
[15] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
[16] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
[17] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
[18] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53–70, 2006. doi:10.2478/v10037-006-0008-x.[Crossref]
[19] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491–494, 2001.
[20] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495–500, 2001.
[21] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167–175, 2008. doi:10.2478/v10037-008-0023-1.[Crossref] | Zbl 1321.46022
[22] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Hopf extension theorem of measure. Formalized Mathematics, 17(2):157–162, 2009. doi:10.2478/v10037-009-0018-6.[Crossref] | Zbl 1276.46015
[23] Noboru Endou, Kazuhisa Nakasho, and Yasunari Shidama. σ-ring and σ-algebra of sets. Formalized Mathematics, 23(1):51–57, 2015. doi:10.2478/forma-2015-0004.[Crossref] | Zbl 1317.28001
[24] P. R. Halmos. Measure Theory. Springer-Verlag, 1974. | Zbl 0283.28001
[25] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573–577, 1997.
[26] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339–345, 1996.
[27] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.
[28] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745–749, 1990.
[29] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.
[30] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449–452, 1991.
[31] M.M. Rao. Measure Theory and Integration. CRC Press, 2nd edition, 2004. | Zbl 1108.28001
[32] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187–190, 1990.
[33] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
[34] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
[35] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
[36] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.