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Weakly mixing transformations and the Carathéodory definition of measurable sets

Amos Koeller, Rodney Nillsen, Graham Williams (2007)

Colloquium Mathematicae

Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 𝕋 given by T(z) = vz. It is known that the problem of calculating the outer measure of a T-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, T is not weakly mixing. Now there is an example, due to Kakutani, of a transformation ψ̃ which is weakly mixing but not strongly...

What’s the price of a nonmeasurable set?

Mirko Sardella, Guido Ziliotti (2002)

Mathematica Bohemica

In this note, we prove that the countable compactness of { 0 , 1 } ℝ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of ℝ . This is done by providing a family of nonmeasurable subsets of ℝ whose intersection with every non-negligible Lebesgue measurable set is still not Lebesgue measurable. We develop this note in three sections: the first presents the main result, the second recalls known results concerning non-Lebesgue measurability and its relations with the Axiom...

When ℵ₁ many sets are contained in a countably generated σ-field

R. Drabiński, E. Grzegorek (2009)

Colloquium Mathematicae

We discuss the problem when ℵ₁ sets are contained in a σ-generated σ-field on some set X. This is related to a problem raised by K. P. S. Bhaskara Rao and Rae Michael Shortt [Dissertationes Math. 372 (1998)] which we answer. We also briefly discuss generating the family of all subsets from rectangles.

When is ℝ the union of an increasing family of null sets?

Juan González-Hernández, Fernando Hernández-Hernández, César E. Villarreal (2007)

Commentationes Mathematicae Universitatis Carolinae

We study the problem in the title and show that it is equivalent to the fact that every set of reals is an increasing union of measurable sets. We also show the relationship of it with Sierpi'nski sets.

σ -porosity is separably determined

Marek Cúth, Martin Rmoutil (2013)

Czechoslovak Mathematical Journal

We prove a separable reduction theorem for σ -porosity of Suslin sets. In particular, if A is a Suslin subset in a Banach space X , then each separable subspace of X can be enlarged to a separable subspace V such that A is σ -porous in X if and only if A ∩ V is σ -porous in V . Such a result is proved for several types of σ -porosity. The proof is done using the method of elementary submodels, hence the results can be combined with other separable reduction theorems. As an application we extend a theorem...

σ-ring and σ-algebra of Sets1

Noboru Endou, Kazuhisa Nakasho, Yasunari Shidama (2015)

Formalized Mathematics

In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18],...

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