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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 countably infinite...
In this paper we present and develop, only elementarily, an axiomatic frame for some special fuzzy relations, the so called relational probabilities, which happen to be families of functions which in some cases are quantic probabilities [1] on sublattices of a given lattice; in this way we obtain calculations similar to the ordinary ones but based on a weaker lattice background. This framework is inspired on the presentation of conditional probability on Boolean algebras made in [5] and in [4] and...
In this paper, the authors introduce the notion of conditional expectation of an observable on a logic with respect to a sublogic, in a state , relative to an element of the logic. This conditional expectation is an analogue of the expectation of an integrable function on a probability space.
The ring of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring of all continuous functions and, similarly, the ring of all Borel measurable subsets of is a sequential ring completion of the subring of all finite unions of half-open intervals; the two completions are not categorical. We study -rings of maps and develop a completion theory covering the two examples. In particular, the -fields of sets form an epireflective...
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