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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.

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Perhaps this could help http://arxiv.org/abs/1205.4504 bye W.

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Gleason theorem (for von Neumann algebras, and even JBW algebras, without type I_2 part) has been extended to signed / complex measures (and even suitable vector valued measures). S.Maeda, Bunce and Maitland wright wrote surveys around 20 years ago. –  user24527 Jun 26 '12 at 23:14

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