MEASURABILITY OF UNIONS OF CERTAIN DENSE SETS

In this paper we study measurability properties of sets of the form E( t) = {t + m alpha(1) + n alpha(2)\m, n epsilon Z} (t epsilon R) where alpha(1), alpha(2) are given real numbers with alpha(1)/alpha(2) irrat ional. Sets such as these have played an important role to establish c ertain 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 O < mA < mT, where m denotes the Lebesgue measure in R and A = boolean OR(t eps ilon C) E(t) boolean AND T. In fact, we prove that if T is any Lebesgu e measurable subset of R, then no such sets can exist: no matter which C we choose, if A is measurable then mA equals O or mT. Moreover, if A is a nonmeasurable set, the same applies to its Lebesgue outer measu re. However, if we remove the condition on T of being measurable, we p rovide an example of (nonmeasurable) sets C and T for which the outer measure of A lies in between O and the outer measure of T.