For any C ⊆ R there is a subset A ⊆ C such that A + A has inner measure zero and outer measure the same as C + C. Also, there is a subset A of the Cantor middle third set such that A+A is Bernstein in [0, 2]. On the other hand there is a perfect set C such that C + C is an interval I and there is no subset A ⊆ C with A + A Bernstein in I.

In this note we will show that for every natural number n > 0 there exists an S ⊂ [0, 1] such that its n-th algebraic sum nS = S + · · ·+ S is a nowhere dense measure zero set, but its n+1-st algebraic sum nS+S is neither measurable nor it has the Baire property. In addition, the set S will be also a Hamel base, that is, a linear base of R over Q. We use the standard notation as in [2]. Thus sy...

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Togethe...

1. Introduction. Let 5 be a measure space with a countably additive , (T-finite, complete (non-negative) measure. J. von Neumann has proved that, from each class x of measurable sets modulo null sets of 5, a representative set R(x)Ex can be picked in such a way