Related papers: Foundations of abstract probability theory
Modeling worlds and actions under uncertainty is one of the central problems in the framework of decision-theoretic planning. The representation must be general enough to capture real-world problems but at the same time it must provide a…
We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures.…
There has not been an established mathematical measure of evidence. Some Bayesians have argued that probability can be an objectively correct measure of ``rational degrees of belief,'' which we do not distinguish from evidence. However,…
We introduce the concept of compact quantitative equational theory. A quantitative equational theory is defined to be compact if all its consequences are derivable by means of finite proofs. We prove that the theory of interpolative…
In this paper we analyse some of the classical paradoxes in Social Choice Theory (namely, the Condorcet paradox, the discursive dilemma, the Ostrogorski paradox and the multiple election paradox) using a general framework for the study of…
Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on F\o lner sequences for amenable groups, entropy defined using a random ordering of…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense,…
We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…
A general theory of summation of divergent series based on the Hardy-Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some…
This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved…
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…
A common meadow is an enrichment of a field with a partial division operation that is made total by assuming that division by zero takes the a default value, a special element $\bot$ adjoined to the field. To a common meadow of real numbers…
This paper covers two topics: first an introduction to Algorithmic Complexity Theory: how it defines probability, some of its characteristic properties and past successful applications. Second, we apply it to problems in A.I. - where it…
The Allais and Ellsberg paradoxes show that the expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions. The popular explanation in terms of 'ambiguity aversion' is not completely accepted. On the…
In this paper we study the connections of three paradigms in number theory: the adelic formulation of the Riemann zeta function, the Weil explicit formula and the concepts of the so called probabilistic number theory initiated by Harald…
This research introduces a new method for the transition from partial to ordinary differential equations that is based on the Kolmogorov superposition theorem. In this paper, we discuss the numerical implementation of the Kolmogorov theorem…
We present the foundational theory of condensed sets and basic condensed algebra after having introduced key concepts from category theory and homological algebra. In the later sections, we indicate the relevance of condensed mathematics to…
Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of…
This paper contains a development of the Theory of Lebesgue and Bochner spaces of summable functions. It represents a synthesis of the results due to H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford, P. Halmos,…