MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

Perhaps this could help bye W.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.