Real Analysis Exchange

Algebraic sums of sets in Marczewski-Burstin algebras.

Francois G. Dorais and Rafał Filipów

Full-text: Open access

Abstract

Using almost-invariant sets, we show that a family of Marczewski--Burstin algebras over groups are not closed under algebraic sums. We also give an application of almost-invariant sets to the difference property in the sense of de~Bruijn. In particular, we show that if $G$ is a perfect Abelian Polish group then there exists a Marczewski null set $A \subseteq G$ such that $A+A$ is not Marczewski measurable, and we show that the family of Marczewski measurable real valued functions defined on $G$ does not have the difference property.

Article information

Source
Real Anal. Exchange Volume 31, Number 1 (2005), 133-142.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516801

Mathematical Reviews number (MathSciNet)
MR2218194

Zentralblatt MATH identifier
1106.28001

Subjects
Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 39A70: Difference operators [See also 47B39]

Keywords
algebraic sum Marczewski--Burstin algebra Marczewski measurable set Miller measurable set perfect set superperfect set almost-invariant set difference property

Citation

Dorais, Francois G.; Filipów, Rafał. Algebraic sums of sets in Marczewski-Burstin algebras. Real Anal. Exchange 31 (2005), no. 1, 133--142.https://projecteuclid.org/euclid.rae/1149516801


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