Orders of Absolute Measurability

A subset A of the torus 0 1 k is called absolute measurable if the value of μ A is the same for every finitely-additive translation-invariant probability measure μ defined on all subsets of 0 1 k We define four set functions (called orders) that measure how “strongly” a set A is absolute measurable. The order o A equals k − dimB ∂A and is connected to the Jordan measurability of A The order δ A measures how small the oscillation of the average of n translates of χA can be. The order τ A is related to the absolute inner and outer measures defined by Tarski; finally, σ A is defined by the oscillation of those functions that are “scissorcongruent” to χA We prove that o δ τ σ that is, each of the orders o δ τ σ is “finer” than the previous one. We investigate the connection between the orders and questions of equidecomposability. We show that, under certain conditions, a set of large order is equidecomposable to a cube and present some results in the other direction as well.