Measurability of unions of certain dense sets

Abstract

In this paper we study measurability properties of sets of the form \[ E t = { t + m α 1 + n α 2 | m , n ∈ Z } ( t ∈ R ) {E_t} = \{ t + m{\alpha _1} + n{\alpha _2}|m,n \in {\mathbf {Z}}\} \quad (t \in {\mathbf {R}}) \] where α 1 , α 2 {\alpha _1},{\alpha _2} are given real numbers with α 1 / α 2 {\alpha _1}/{\alpha _2} irrational. Sets such as these have played an important role to establish certain fundamental results in measure theory. However, the question of measurability of unions of these sets seems not to have been solved. In an initial guess, no sets C and T seem apparent for which 0 > m A > m T 0 > mA > mT , where m denotes the Lebesgue measure in R and A = ∪ t ∈ C E t ∩ T A = { \cup _{t \in C}}{E_t} \cap T . In fact, we prove that if T is any Lebesgue measurable subset of R, then no such sets can exist: no matter which C we choose, if A is measurable then mA equals 0 or mT. Moreover, if A is a nonmeasurable set, the same applies to its Lebesgue outer measure. However, if we remove the condition on T of being measurable, we provide an example of (nonmeasurable) sets C and T for which the outer measure of A lies in between 0 and the outer measure of T.