The extent of non-conglomerability of finitely additive probabilities

Summary An arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition. If the probability is defined on a -field, the bound is sharp. Hence, non-conglomerability (or equivalently non-disintegrability) characterizes finitely as opposed to countably additive probability. Nonetheless, there exists a PFA probability which is simultaneously conglomerable over an arbitrary finite set of partitions.Neither conglomerability nor non-conglomerability in a given partition is closed under convex combinations. But the convex combination of PFA ultrafilter probabilities, each of which cannot be made conglomerable in a common margin, is singular with respect to any finitely additive probability that is conglomerable in that margin.     

Extensions of finitely additive measures

Translated from Matematicheskie Zametki, Vol. 48, No. 1, pp. 56–60, July, 1990.     

Pairs of tiles which admit finitely or countably infinitely many tilings

For every positive integer r, there exist pairs of prototiles which admit exactly r distinct tilings of the plane. Furthermore, there exist pairs of prototiles which admit a countable infinity of distinct tilings.     

Which families of finite-dimensional joint distributions are associated with continuous-path,countably additive,stochastic processes?

Summary Necessary and sufficient conditions on a family of finite-dimensional distributions for it to be the family of finite-dimensional joint distributions of a countably additive stochastic process with continuous paths have been given by Bartoszynski in 1962. The present paper gives such conditions when the parameter space is any compact metric space and, in particular, any compact manifold. Entrata in Redazione il 23 aprile 1976. This research was prepared with the support of National Science Foundation Grant Nos. MPS 75-09459 and GP 43085.     

A characterization of the families of finite-dimensional distributions associated with countably additive stochastic processes whose sample paths are inD

Summary Associated with any countably additive probability measureP on the well-known Skorohod spaceD is the family {P } of P's finite-dimensional distributions. This paper characterizes all such {P }.This research was prepared with the support of National Science Foundation Grant No. MPS75-09459     

Extremal extension of a finitely additive measure

Translated from Matematicheskie Zametki, Vol. 43, No. 1, pp. 25–30, January, 1988.     

Extensions of semigroup valued,finitely additive measures

LetC be a field of subsets of a non-empty setX and let μ:C→E be a finitely additive measure (a “charge”) taking values in a commutative semigroupE. We consider the problem of extending μ to a charge defined on the power set and we say thatE has the charge extension property (CEP) if such extensions always exist. Los and Marczewski proved [4] that the semigroup of non-negative reals has CEP, and Carlson and Prikry [2] have shown that everygroup has CEP. We prove that every compact semigroup has CEP and show that CEP follows from certain completeness and distributivity conditions. Specializing to the case of lattices considered as semigroups under the operation of supremum, we characterize the class of lattices with CEP. An application to closure operators in general topology is also discussed.     

Elementary equivalence of rings with finitely generated additive groups

We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.     

Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)     

Some finitely additive versions of the strong law of large numbers

LetX be a non-empty set,H= X{su\t8, \gs = \lj{in1}x\lj{in2}x,σ=γ ₁×γ ₂×… be an independent strategy onH, and {Y _n} be a sequence of coordinate mappings onH. The following strong law in a finitely additive setting is proved: For some constantr≧1, if \GS _n=1 ^\t8 {\GS(\vbY _n \vb^2r )n ¹⁺ⁿ < \t8 andσ(Y _n)=0 for alln=1, 2, …, then \1n\gS{inj-1}/{sun} Y{inj}Y _jconverges to 0 withσ-measure 1 asn → ∞. The theorem is a generalization of Chung’s strong law in a coordinate representation process. Finally, Kolmogorov’s strong law in a finitely additive setting is proved by an application of the theorem. This research was based in part on the author’s doctoral dissertation submitted to the University of Minnesota, and was written with the partial support of the United States Army Grant DA-ARO-D-31-124-70-G-102.     

On the use of purely finitely additive multipliers in mathematical programming

In this paper, duality results are obtained for the problem of finding =inf{f(x):g(x)0,g(x)Y,xC}, where the constrainst spaceY is someL -space, by using the norm topology ofY. The corresponding multiplier is the sum of a countably additive part and a purely finitely additive part. Conditions are given such that the purely finitely additive part may be discarded.     

Loeb measure,finitely additive measures,and a theorem of Banach

The Loeb measure construction from nonstandard analysis is applied to two theorems in standard measure theory. In both cases the essential simplification offered by the approach is the ability to work with a σ-additive measure space, even if the hypotheses only guarantee finite additivity. The key to this simplification is the property of -saturated nonstandard models, that any finitely additive measure on an internal algebra extends immediately to a σ-additive measure.      

Loeb measure, finitely additive measures, and a theorem of Banach

The Loeb measure construction from nonstandard analysis is applied to two theorems in standard measure theory. In both cases the essential simplification offered by the approach is the ability to work with a σ-additive measure space, even if the hypotheses only guarantee finite additivity. The key to this simplification is the property of à₁{\aleph_1}-saturated nonstandard models, that any finitely additive measure on an internal algebra extends immediately to a σ-additive measure.     

A finitely additive version of kolmogorov’s law of the iterated logarithm

In their paperSome finitely additive probability (to appear in Ann. Probability), Roger A. Purves and William D. Sudderth introduced the measurable strategy idea. In this paper, we first generalize the measurable strategy idea to the more general sigma-fields of subsets ofX and prove an important theorem. Then, based on this theorem, we state and prove a finitely additive version of Kolmogorov’s law of the iterated logarithm and a finitely additive version of Hartman and Wintner’s law of the iterated logarithm in a finitely additive setting. This research was written with the partial support of the U.S. Army Grant DA-ARO-D-31-124-70-G-102.