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 realvalued function on the 2sphere in 3space 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.
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Perhaps this could help http://arxiv.org/abs/1205.4504 bye W. 

