Abstract
For a vector lattice E with the principal projection property, the following generalization of H.Freudenthal's spectral theorem is proved: There exists a measure space (Ω,R,π) such that integration with respect to π establishes a vector lattice isomorphism from L1(π) to E. Here π:ℛ→E is a σ -additive vector measure on some δ-ring R which, for [σ-] Dedekind complete E, may be chosen to be the δ-ring of relatively compact [Baire-] Borel sets in a locally compact space.
Among others Kakutani's representation of abstract L-spaces as concrete L1 -spaces is an immediate consequence.
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Hackenbroch, W. Über den Freudenthalschen Spektralsatz. Manuscripta Math 13, 83–99 (1974). https://doi.org/10.1007/BF01168745
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DOI: https://doi.org/10.1007/BF01168745