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Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

Jack Brown, Hussain Elalaoui-Talibi (1999)

Colloquium Mathematicae

ℒ denotes the Lebesgue measurable subsets of ℝ and ℒ 0 denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 h a s a p e r f e c t s u b s e t Q ∈ ℒ $ ℒ 0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ℒ 0 ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal ( s 0 ) which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

Measurable cardinals and category bases

Andrzej Szymański (1991)

Commentationes Mathematicae Universitatis Carolinae

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

Measure-theoretic unfriendly colorings

Clinton T. Conley (2014)

Fundamenta Mathematicae

We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measure-preserving actions of groups with designated finite generating set. As a corollary, we obtain the existence of translation-invariant random unfriendly colorings...

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