Gleason's theorem (Journal of Mathematics and Mechanics, Vol. 6, No. 6, 1957) classifies measures on the closed subspaces of a separable Hilbert space. A key lemma toward the proof of the theorem asserts the following. Let f be a nonnegative real-valued function on the 2-sphere in 3-space with the property that its sum on any triple of orthogonal points is independent of the triple. Then f is continuous. The question is whether the conclusion remains valid if one drops the nonnegativity assumption and
assumes Borel measurability instead.