We say that agreeing to agree is possible for an event E if there exist posterior beliefs of the agents with a common prior such that it is common knowledge that the agents’ posteriors for E coincide. We propose a notion called Dutch book which is a profile of interim contracts between an outsider and the agents based on the occurrence of E, such that the outsider makes positive profit in all s...

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those ...

Prompted by a recent question of G. Hjorth [12] as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we answer this question and more generally investigate partitions of countable metric spaces. We show that an indivisible metric space must be totally Cantor...

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s approach to partition, and it is motivated by Seward’s recent generalization Rokhlin’s characterization from amenable general groups. A key ingredient in our the use an auxiliary probability-meas...

This paper focuses on completeness results about generic expansions of logics of both continuous t-norms and Weak Nilpotent Minimum (WNM) with truth-constants. Indeed, we consider algebraic semantics for expansions of these logics with a set of truth-constants {r | r ∈ C}, for a suitable countable C ⊆ [0, 1], and provide a full description of completeness results when (i) either the t-norm is a...

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational princi...

For a continuous action of a countable discrete group G on a Polish space X, a countable Borel partition P of X is called a generator if GP ∶= {gP ∶ g ∈ G,P ∈ P} generates the Borel σ-algebra of X. For G = Z, the Kolmogorov–Sinai theorem gives a measuretheoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite...

The following theorem is proved, answering a question raised by Davies in 1963. If L0 ∪L1 ∪L2 ∪ . . . is a partition of the set of lines of R, then there is a partition R = S0 ∪ S1 ∪ S2 ∪ . . . such that |` ∩ Si| ≤ 2 whenever ` ∈ Li. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson & Mauldin. 0. Introduction. A series of papers, b...