Borel classes in AST. Measurability, cuts and equivalence
Borel equivalence and isomorphism of coanalytic sets [Book]
Borel extensions of Baire measures
We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom ♣ there is a Dowker space which is quasi-Mařík but not Mařík, answering a question of H. Ohta and K. Tamano, and under P(c), that there is a Mařík Dowker space, answering a question of W. Adamski. We answer further questions of H. Ohta and K. Tamano by showing that the union of a Mařík space and a compact space is Mařík,...
Borel extensions of Baire measures in ZFC
We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.
Borel matrix
We study the Borel summation method. We obtain a general sufficient condition for a given matrix to have the Borel property. We deduce as corollaries, earlier results obtained by G. M“uller and J.D. Hill. Our result is expressed in terms belonging to the theory of Gaussian processes. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical systems. However, in the -setting, we establish necessary conditions of the same kind by using Bourgain’s...
Borel partitions of unity and lower Carathéodory multifunctions
We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations...
Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space
For a linear operator T in a Banach space let denote the point spectrum of T, let for finite n > 0 be the set of all such that dim ker(T - λ) = n and let be the set of all for which ker(T - λ) is infinite-dimensional. It is shown that is , is and for each finite n the set is the intersection of an set and a set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more detailed decomposition...
Borel semigroups and probabilities.
Borel sets of exact class
Borel sets with σ-compact sections for nonseparable spaces
We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.
Borel structures for function spaces
Borel-approachable functions
Borel-dense Blackwell spaces are strongly Blackwell
Bounded analytic sets in Banach spaces
Conditions are given which enable or disable a complex space to be mapped biholomorphically onto a bounded closed analytic subset of a Banach space. They involve on the one hand the Radon-Nikodym property and on the other hand the completeness of the Caratheodory metric of .
Bounded convergence theorem and integral operator for operator valued measures
Bounded linear functionals on the space of Henstock-Kurzweil integrable functions
Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.
Bounded variation and invariant measures
Boundedly expressible sets
For a given sequence a boundedly expressible set is introduced. Three criteria concerning the Hausdorff dimension of such sets are proved.
Boundedness conditions of Hausdorff -measure in metric spaces.
Brownian motion, approximation of functions, and Fourier analysis