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

Formalized Mathematics (2014)

• Volume: 22, Issue: 3, page 199-204
• ISSN: 1426-2630

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## Abstract

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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).

## How to cite

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Peter Jaeger. "Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance." Formalized Mathematics 22.3 (2014): 199-204. <http://eudml.org/doc/270819>.

@article{PeterJaeger2014,
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).},
author = {Peter Jaeger},
journal = {Formalized Mathematics},
keywords = {event; Borel set; random variable},
language = {eng},
number = {3},
pages = {199-204},
title = {Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance},
url = {http://eudml.org/doc/270819},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Peter Jaeger
TI - Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 3
SP - 199
EP - 204
AB - 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).
LA - eng
KW - event; Borel set; random variable
UR - http://eudml.org/doc/270819
ER -

## References

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