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Random Variables and Product of Probability Spaces

Hiroyuki Okazaki, Yasunari Shidama (2013)

Formalized Mathematics

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 countably infinite...

Relative conditional expectations on a logic

Oľga Nánásiová, Sylvia Pulmannová (1985)

Aplikace matematiky

In this paper, the authors introduce the notion of conditional expectation of an observable x on a logic with respect to a sublogic, in a state m , relative to an element a of the logic. This conditional expectation is an analogue of the expectation of an integrable function on a probability space.

Rings of maps: sequential convergence and completion

Roman Frič (1999)

Czechoslovak Mathematical Journal

The ring B ( R ) of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring C ( R ) of all continuous functions and, similarly, the ring 𝔹 of all Borel measurable subsets of R is a sequential ring completion of the subring 𝔹 0 of all finite unions of half-open intervals; the two completions are not categorical. We study 0 * -rings of maps and develop a completion theory covering the two examples. In particular, the σ -fields of sets form an epireflective...

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