Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the twenty years since its creation, fr...

In probability theory, we are given a set Ω of outcomes ω, and try to consider the probabilities of subsets A ⊂ Ω, which we call events. To this end, we define a probability measure Pr(·) or μ(·) (we will use each notation interchangably) that is a function from these sets A ⊂ Ω → [0, 1]. As a technical complicate, μ or Pr may only be defined on some of the subsets of Ω, but this will be adress...

1 Brownian Motion 2 1.

I Axioms of Probability 2 I-A Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-A1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-A1a Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-A1b Mutually Exclusive Sets or Disjoint Sets . . . . . . . . . . . . . . . . . . . 2...

In this paper, five conditions that have been proposed by Cobb and Shenoy are studied for nine different mappings from the Dempster-Shafer theory to the probability theory. After comparing these mappings, one of the considerable results indicates that none of the mappings satisfies the condition of invariance with respect to the marginalization process. In more details, the main reason for this...