Splitting of liftings in products of probability spaces II

For a probability measure R on a product of two probability spaces that is absolutely continuous with respect to the product measure we prove the existence of liftings subordinated to a regular conditional probability and the existence of a lifting for R with lifted sections which satisfies in addition a rectangle formula. These results improve essentially some of the results from the former work of the authors [W. Strauss, N.D. Macheras, K. Musia?, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389-2408], by weakening considerably the assumptions and by presenting more direct and shorter proofs. In comparison with [W. Strauss, N.D. Macheras, K. Musia?, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389-2408] it is crucial for applications intended that we can now prescribe one of the factor liftings completely freely. We demonstrate the latter by applications to τ-additive measures, transfer of strong liftings, and stochastic processes.     

Weakly compact operators on non-complete normed spaces

We prove that weakly compact operators on a non-reflexive normed space cannot be bijective. We also show that, in the above result, bijectivity cannot be relaxed to surjectivity. Finally, we study the behaviour of surjective weakly compact operators on a non-reflexive normed space, when they are perturbed by small scalar multiples of the identity, and derive from this study the recent result of Spurný [A note on compact operators on normed linear spaces, Expo. Math. 25 (2007) 261–263] that compact operators on an infinite-dimensional normed space cannot be surjective.     

Hurwitz spaces and liftings to the Valentiner group

We study the components of the Hurwitz scheme of ramified coverings of ${\mathbb{P}}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the 3-cycles, we use as invariant the lifting to the Valentiner group, a triple covering of $A_6$. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution found by Bogomolov and Kulikov.     

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

Measures on product spaces and the existence of strong Baire liftings

Some notions are introduced for studying measures on product spaces, the main concept being that of property (*). In case when the topological factors are separable metric spaces, this property is equivalent to the completion regularity. We prove that (*) is preserved under arbitrary products of measure spaces. As a consequence, we deduce a series of related results in measure theory (some of which are known). In particular, the following extension of a result by Losert is obtained: Subject to CH, every product of ₂ many completion regular measures, each supported on any product of ₁ many compact metric spaces admits a strong Baire lifting.     

An infinitary probability logic for type spaces

Type spaces in the sense of Harsanyi (1967/68) play an important role in the theory of games of incomplete information. They can be considered as the probabilistic analog of Kripke structures. By an infinitary propositional language with additional operators “individual i assigns probability at least α to” and infinitary inference rules, we axiomatize the class of (Harsanyi) type spaces. We prove that our axiom system is strongly sound and strongly complete. To the best of our knowledge, this is the very first strong completeness theorem for a probability logic with σ-additive probabilities. We show this by constructing a canonical type space whose states consist of all maximal consistent sets of formulas. Furthermore, we show that this canonical space is universal (i.e., a terminal object in the category of type spaces) and beliefs complete.     

Computable randomness and betting for computable probability spaces

Unlike Martin‐Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting “bit‐wise” definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness—endomorphism randomness—which the author hopes will shed light on the open question of whether Kolmogorov‐Loveland randomness is equivalent to Martin‐Löf randomness. The last section contains ideas for future research.     

Duality and perfect probability spaces

Given probability spaces let denote the set of all probabilities on the product space with marginals and and let be a measurable function on Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubin\v{s}tein (1958) for the case of compact metric spaces are concerned with the validity of the duality

(where is the collection of all probability measures on with and as the marginals). A recently established general duality theorem asserts the validity of the above duality whenever at least one of the marginals is a perfect probability space. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterization of perfect probability spaces.

     

A remark on probability spaces

The object of this paper is to exhibit some aspects of Kolmogorov's axiomatic theory of probability related to its phenomenological interpretation.J. P. Gabriel has informed the editor that P. Milasevic was killed in a mountain accident on October 27, 1989.     

Isoperimetric inequality for radial probability measures on Euclidean spaces

We generalize the Poincaré limit which asserts that the n-dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first n-coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps. Using this generalization, we derive a Gaussian isoperimetric inequality for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.     

A dichotomy for the convex spaces of probability measures

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has ‘small’ local character in M or else M contains a measure of ‘large’ Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.     

Idempotent probability measures on ultrametric spaces

Following the construction due to Hartog and Vink we introduce a metric on the set of idempotent probability measures (Maslov measures) defined on an ultrametric space. This construction determines a functor on the category of ultrametric spaces and nonexpanding maps. We prove that this functor is the functorial part of a monad on this category. This monad turns out to contain the hyperspace monad.