This note concerns subsets A of the unit 2-sphere 5 such that (*) for each continuous real-valued mapping/ of 5 there exists a rotation r of S with all points of r(A) having the same value under/. In 1942, Kakutani [3 ] proved that the set A of end points of an orthonormal set of 3 vectors has property (*). It was observed by de Mira Fernandes [5 ] that the same proof holds in case A is the set...

Solovay's random-real forcing ([1]) is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay's model that do not follow from the existence of real-valued measurability.