Given an action of a group $\Gamma$ on a measure space $\Omega$, we provide a sufficient criterion under which two sets $A, B\subseteq \Omega$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $\Gamma$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $n\ge 3$, with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension $n\ge 2$.

/pdf/measurable-equidecompositions-for-group-actions-with-an-3yrbmlk2db.pdf

Measurable equidecompositions for group actions with an expansion property

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure—category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the preface of Kelley (General Topology, Van Nostrand, Toronto, 1995): “what every young analyst should know”.

/pdf/set-theory-and-the-analyst-5dydntdyu6.pdf

Set theory and the analyst

We show that several theorems about Polish spaces, which depend on the axiom of choice (
 $$\mathcal {AC}$$
 ), have interesting corollaries that are theorems of the theory $$\mathcal {ZF} + \mathcal {DC}$$
 , where $$\mathcal {DC}$$
 is the axiom of dependent choices. Surprisingly it is natural to use the full $$\mathcal {AC}$$
 to prove the existence of these proofs; in fact we do not even know the proofs in $$\mathcal {ZF} + \mathcal {DC}$$
 . Let $$\mathcal {AD}$$
 denote the axiom of determinacy. We show also, in the theory $$\mathcal {ZF} + \mathcal {AD} + V = L(\mathbb {R})$$
 , a theorem which strenghtens and generalizes the theorem of Drinfeld (Funct Anal Appl 18:245–246, 1985) and Margulis (Monatshefte Math 90:233–235, 1980) about the unicity of Lebesgue’s measure. This generalization is false in $$\mathcal {ZFC}$$
 , but assuming the existence of large enough cardinals it is true in the model $$\langle L(\mathbb {R}),\in \rangle $$
 .

Shadows of the axiom of choice in the universe $$L(\mathbb {R})$$ L ( R )

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure -- category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know".

Set Theory and the Analyst

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

Variants on the Berz sublinearity theorem

We study a property of smallness of sets which is stronger than the possibility of packing the set into arbitrarily small balls (i.e., being Tarski null) but weaker than paradoxical decomposability (i.e., being a disjoint union of two sets equivalent by finite decomposition to the whole). We show, using the Axiom of Choice for uncountable families, that there are Tarski null sets which are not small sets. Using only the Principle of Dependent Choices, we show that bounded subsets of R that are included in countable unions of proper analytic subsets of R are small, and several related results.

/pdf/on-small-subsets-in-euclidean-spaces-53pi12igus.pdf

On Small Subsets in Euclidean Spaces

A set $$E \subset \mathbb {R}^n$$
 is called uniformly discrete if there exists an $$\varepsilon >0 $$
 such that no two points of E are closer than $$\varepsilon $$
 . Applying a theorem of T. Tao on the absence of paradoxical decompositions of uniformly discrete sets we will prove, under an additional assumption, that such a set $$E \subset \mathbb {R}^n$$
 has at most one point p such that $$E {\setminus } \{p\}$$
 and E are congruent. We prove also that if $$E \subseteq \mathbb {R}^n$$
 is a discrete set and G is a discrete subgroup of the group of isometries of $$\mathbb {R}^n$$
 then there is at most one point $$p \in E$$
 such that there exists a $$\varphi \in G$$
 with $$\varphi (E) = E {\setminus } \{p\}$$
 . Related unsolved problems will be pointed out.

Paradoxical sets and sets with two removable points

/pdf/intersection-of-generic-rotations-in-some-classical-spaces-2a303vpz66.pdf

Grzegorz Tomkowicz

Papers

Shadows of the axiom of choice in the universe $$L(\mathbb {R})$$ L ( R )

On Small Subsets in Euclidean Spaces

Paradoxical sets and sets with two removable points

Intersection of Generic Rotations in Some Classical Spaces

A measure theoretic paradox from a continuous colouring rule