Orders of absolute measurability
The author considers absolutely measurable subsets of the torus \([0,1)^k\) and various questions concerning equidecomposability of such sets. \(A\) is absolutely measurable if the value of \(\mu(A)\) is the same for every Banach measure \(\mu\) (i.e., a finitely additive and translation-invariant probability measure defined on all subsets of \([0,1)^k\)). He defines four set-functions (called orders): the Jordan order \(o(A)\), the absolute order \(\delta(A)\), the Tarski order \(\tau(A)\) and the scissor order \(\sigma(A)\), and shows that for every set \(A\subset[0,1)^k\): \(o(A)\ll \delta(A)\ll\tau(a)\ll\sigma(A)\). (Here \(\alpha\ll\beta\) means that there is a positive constant \(c\) only depending on \(k\) such that \(\alpha\leq c\cdot\beta\).) Finally, the author establishes relationships between orders of a set \(A\) and equivalence of \(A\) to a cube. For example: If \(A\) is equivalent to a cube then \(\delta(A)>0\) and \(\tau(A)=\sigma(A)=\infty\). If \(\lambda_k(A)>0\) and \(o(A)>0\) then \(A\) is equivalent to a cube. If \(\delta(A)=\infty\) then \(A\) is equivalent to a cube.
scientific article; zbMATH DE number 1545041