<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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).</p></abstract>

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).

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

Siegmund-Schacky-Str. 18a 80993 Munich, Germany

Formalized Mathematics

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

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...

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

Published Online: Mar 31, 2014

Page range: 199 - 204

Received: Jul 10, 2014

DOI: https://doi.org/10.2478/forma-2014-0022

© by Peter Jaeger

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