On ergodic quasi-invariant measures of group automorphism

We study the dynamics of projective transformations and apply it to (i) prove that the isotropy subgroups of probability measures on algebraic homogeneous spaces are algebraic and to (ii) study the class of ergodic quasi-invariant measures of automorphisms of non-compact Lie groups. It is shown that their support is always a proper subset and that under certain conditions on the Lie group the induced homeomorphism of the support is topologically equivalent to a translation of a compact group.     

On invariant measures and ergodic theorems for positive operators

Ergodic theorems and the existence of a finite invariant measure are investigated for a positive linear operator T on L₁ of a σ-finite measure satisfying sup_{N ? 1}∥N^?1Σ_{n = 0}^{N ? 1}Tⁿ∥< ∞.     

On multifunction transforms of probability measures

The paper briefly presents Topsøe's setup and gives two results based on that. A delta theorem for multifunctions improving that of King [2] is given. In particular, the contingent derivative may be a compact-valued instead of a single-valued multi-function. Stochastic optimization is also discussed. Some results are achieved for the limit distribution of optimal values as well as of optimal solutions.     

On semi-invariant probability measures on semigroups

Summary Let a regular Borel measure m on a locally compact semigroup S be upper semi-invariant i.e., m(C x)m(C) and m(x C)m(C) for every compact C and x in S. It is shown: (i) Every subsemigroup of S of positive measure contains an idempotent. (ii) S admits an upper semi-invariant probability measure iff S has a kernel K which is a compact group.We should like to thank the referee for pointing out certain redundancies in the theorems. Also we thank Dr. Tze-Chien Sun for some helpful observations.     

On measures ergodic with respect to an analytic equivalence relation

In this paper, we prove that the set of probability measures which are ergodic with respect to an analytic equivalence relation is an analytic set. This is obtained by approximating analytic equivalence relations by measures, and is used to give an elementary proof of an ergodic decomposition theorem of Kechris.

     

On the pointwise ergodic behaviour of transformations preserving infinite measures

We consider situations in which the asymptotic type of a measure preserving transformation manifests itself in a pointwise manner.     

On the unimodality of discrete probability measures

This work was supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec, Canada     

On the existence of invariant probability measures

Let (X,A) be a measureable space andT:X →X a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA ₀⊂A is a convergence class for ℳ such that, for everyA ∈A ₀, the sequence ((1/n) Σ _i ^=0/n−1 1 _A ∘T ⁱ) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.     

On Borel probability measures and noncooperative game theory

In this article, the well-known minimax theorems of Wald, Ville and von Neumann are generalized under weaker topological conditions on the payoff function f and/or extended to the larger set of the Borel probability measures, instead of the set of mixed strategies.     

On the Bayes formula for fuzzy probability measures

The fuzzy P-measure is presented here as the unique useful fuzzy probability measure satisfying the Bayes formula.     

On contiguity of probability measures corresponding to semimartingales

, (ⁿ), - (P ⁿ ), P ⁿ (A ⁿ )>0P ⁿ (A ⁿ )0,n. [15] - , . , P ⁿ P ⁿ T ⁿ T ⁿ .     

The individual ergodic theorem for non-invariant measures

Probability Theory and Related Fields -     

Dichotomy of ergodic measures on linear spaces

Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 942–944, December, 1995.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).