Displaying similar documents to “Standard representation of multivariate functions on a general probability space.”

Random Variables and Product of Probability Spaces

Hiroyuki Okazaki, Yasunari Shidama (2013)

Formalized Mathematics

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We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of...

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

Peter Jaeger (2014)

Formalized Mathematics

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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,...