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Formalized Mathematics

(a computer assisted approach)

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Volume 22, Issue 3


Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance

Peter Jaeger
Published Online: 2014-03-31 | DOI: https://doi.org/10.2478/forma-2014-0022


We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets [16] (pp. 9-10). Literature [16] (pp. 9-10) and [11] (pp. 11-12) gives an overview, that there exists some other sets for this construction. Last we define special functions as random variables for stochastic finance in discrete time. The relevant functions are implemented in the article [15], see [9] (p. 4). The aim is to construct events and random variables, which can easily be used with a probability measure. See as an example theorems (10) and (14) in [20]. Then the formalization is more similar to the presentation used in the book [9]. As a background, further literatures is [3] (pp. 9-12), [13] (pp. 17-20), and [8] (pp.32-35).

Keywords : event; Borel set; random variable


  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [3] Siegfried Bosch. Lineare Algebra. Springer, Berlin, Heidelberg, 4 edition, 2008.Google Scholar

  • [4] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [6] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [7] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [8] Gerd Fischer. Lineare Algebra. Vieweg, Braunschweig, Wiesbaden, 13 edition, 2002.Google Scholar

  • [9] Hans F¨ollmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Google Scholar

  • [10] Otto Forster. Analysis 1. Vieweg-Verlag, Braunschweig/Wiesbaden, 6th edition, 2001. Google Scholar

  • [11] Hans-Otto Georgii. Stochastik, Einf¨uhrung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2nd edition, 2004.Google Scholar

  • [12] Adam Grabowski. On the subcontinua of a real line. Formalized Mathematics, 11(3): 313-322, 2003.Google Scholar

  • [13] Harro Heuser. Lehrbuch der Analysis. Teil 1. Teubner, Stuttgart, Leipzig, Wiesbaden, 15 edition, 2003.Google Scholar

  • [14] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990.Google Scholar

  • [15] Peter Jaeger. Elementary introduction to stochastic finance in discrete time. Formalized Mathematics, 20(1):1-5, 2012. doi:10.2478/v10037-012-0001-5.CrossrefGoogle Scholar

  • [16] Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006.Google Scholar

  • [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Google Scholar

  • [18] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Google Scholar

  • [19] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Google Scholar

  • [20] Hiroyuki Okazaki and Yasunari Shidama. Random variables and product of probability spaces. Formalized Mathematics, 21(1):33-39, 2013. doi:10.2478/forma-2013-0003.CrossrefGoogle Scholar

  • [21] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Google Scholar

  • [22] Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Google Scholar

  • [23] Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Google Scholar

  • [24] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Google Scholar

  • [25] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Google Scholar

  • [26] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.Google Scholar

  • [27] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Google Scholar

  • [28] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [29] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Google Scholar

  • [30] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Set sequences and monotone class. Formalized Mathematics, 13(4):435-441, 2005. Google Scholar

About the article

Received: 2014-07-10

Published Online: 2014-03-31

Published in Print: 2014-09-01

Citation Information: Formalized Mathematics, Volume 22, Issue 3, Pages 199–204, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2014-0022.

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© by Peter Jaeger. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0) BY-SA 3.0

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