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Preliminaries
Martin Buntinas, Loyola University, Chicago
Book:

Classical and Discrete Functional Analysis with Measure Theory

Published online:

06 January 2022

Print publication:

20 January 2022, pp 1-62
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Summary

This preliminary chapter contains notations, definitions, and basic concepts needed for the study of Measure Theory and Functional Analysis. Most of this chapter is for reference and may be read only as needed. Included are concepts such as Convergence, Continuity, and Compactness in Euclidean Spaces. The theory of Euclidean Measure and sets of Measure Zero are covered. An overview is included of Integration sufficient to begin the study of Functional Analysis. The chapter finishes with topics such as Functions of Bounded Variation, Inequalities, along with a discussion of the Axiom of Choice.

Extending Baire property by uncountably many sets
Paweł Kawa, Janusz Pawlikowski
Journal:

The Journal of Symbolic Logic / Volume 75 / Issue 3 / September 2010

Published online by Cambridge University Press:

12 March 2014, pp. 896-904

Print publication:

September 2010
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We show that for an uncountable κ in a suitable Cohen real model for any family {Av}v<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets Av, into the algebra of Baire subsets of 2κ modulo meager sets such that for all Borel B,
The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.

Combinatorial properties of filters and open covers for sets of real numbers
Claude Laflamme, Marion Scheepers
Journal:

The Journal of Symbolic Logic / Volume 64 / Issue 3 / September 1999

Published online by Cambridge University Press:

12 March 2014, pp. 1243-1260

Print publication:

September 1999
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We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set of real numbers have the covering property of Menger if, and only if, each filter on ω associated with its countable ω-cover is a P+ filter.