ACFA and measurability

We show that definable sets of finite S₁-rank in algebraically closed fields with an automorphism can be measured.     

Symmetry and measurability

In the Hewitt–Savage 0-1 law, symmetric measurable sets are considered in a countable product of a σ-algebra Σ with itself. Therefore, it may be of interest to find simple generators for these sets in terms of the generators of Σ. We put the problem into a more general framework of action groups and as an application find simple generators for the finite product case. We also have a partial result in the countable product case.     

Generalized Silver and Miller measurability

We present some results about the burgeoning research area concerning set theory of the “κ‐reals”. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.     

Abstract Riemann integrability and measurability

We prove that the spectral sets of any positive abstract Riemann integrable function are measurable but (at most) a countable amount of them. In addition, the integral of such a function can be computed as an improper classical Riemann integral of the measures of its spectral sets under some weak continuity conditions which in fact characterize the integral representation.     

Uniform unfolding and analytic measurability

We generalize Solovay's unfolding technique for infinite games and use an Unfolding Theorem to give a uniform method to prove that all analytic sets are in the -algebras of measurability connected with well-known forcing notions. Received: 19 July 1996     

Models of Cohen measurability

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.     

Christensen measurability and some functional equation

Let X be a real separable F-space. We characterize solutions ${f:X\to\mathbb{R}}$ and ${M:\mathbb{R}\to\mathbb{R}}$ of the equation f(x?+?M(f(x))y)?=?f(x)f(y) such that f is bounded on a nonzero Christensen measurable set. Our result generalizes [Jab?o??ska in Acta Math Hung 125(1?C2):113?C119 2009, Theorem 1].     

Successors of singular cardinals and measurability

It is shown, starting from a model in which κ < λ, κ is 2^λ supercompact, and λ is a measurable cardinal, how to force and obtain a model in which the Axiom of Choice is false and in which the successor of a singular cardinal is measurable.     

On the notion of measurability

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I wish to record my thanks to Dr. Kummer for his constant encouragement and many helpful suggestions in preparing this paper.     

On measurability of fuzzy-number-valued functions

This paper gives a new definition of the measurability of fuzzy-number-valued functions, and shows the relationship among this measurability and those derived from the corresponding set-valued functions.     

The strength of measurability hypotheses

We prove some consequences of various measurability hypotheses. Especially, we establish that the measurability of Σ ₂ ¹ sets implies that Σ ₂ ¹ sets have the property of Baire.     

Continuity and measurability of multiplier and projective representations

It is shown that various kinds of measurability of a multiplier representation of a locally compact group are equivalent, and are equivalent to the continuity of an ordinary representation of a related group. A unitary representation with measurable coefficients is the sum of a strongly continuous representation and a representation all of whose coefficients are locally null functions. Similar results are obtained for projective unitary representations. This work is carried out without any separability assumptions.     

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single‐valued as well as for multi‐valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya‐Srivastava in order to prove a generalization of the Representation Theorem of Kreitz‐Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach‐Hausdorff‐Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Level by level inequivalence beyond measurability

We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide.     

Aspects of strong compactness, measurability, and indestructibility

 We prove three theorems concerning Laver indestructibility, strong compactness, and measurability. We then state some related open questions. Received: 11 December 1999 / Revised version: 17 September 2000 / Published online: 2 September 2002 Mathematics Subject classification (2000): 03E35, 03E55 Keywords or phrases: Measurable cardinal – Strongly compact cardinal – Supercompact cardinal – Indestructibility