Abstract
In his investigations of the mathematical foundations of quantum mechanics, Mackey1 has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on the closed subspaces means a function μ which assigns to every closed subspace a non-negative real number such that if {A i} is a countable collection of mutually orthogonal subspaces having closed linear span B, then
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The author has been partially supported by the Office of Ordnance Research, Contract No. DA 19-020-ORD-3778. He is indebted also to R. V. Kadison for helpful remarks.
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Notes
See his forthcoming article, ‘Quantum Mechanics and Hilbert Space’, Amer. Math. Monthly.
The present version of this lemma and its proof are due to R. S. Palais, who was kind enough to read the first draft of this paper.
This lemma is due to Jordan and von Neumann, On inner products in linear metric space, Annals of Math. 36 (1935), pp. 719–723.
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© 1975 D. Reidel Publishing Company, Dordrecht, Holland
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Gleason, A.M. (1975). Measures on the Closed Subspaces of a Hilbert Space. In: Hooker, C.A. (eds) The Logico-Algebraic Approach to Quantum Mechanics. The University of Western Ontario Series in Philosophy of Science, vol 5a. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1795-4_7
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DOI: https://doi.org/10.1007/978-94-010-1795-4_7
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