A geometric characterization of non-atomic measure spaces
نویسندگان
چکیده
منابع مشابه
A Banach Space Characterization of Purely Atomic Measure Spaces
It is well known [4, p. 265; 3] that the space 7,i[0, l] is not isomorphic with a conjugate space. At the other extreme, it is also well known that h is isometric with the conjugate space of Co. Each of these is an example of a space of all real-valued integrable functions over a measure space (T, p), a major difference between them being that the measure space underlying 7_i[0, l] has no atoms...
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Let (Ω,Σ, μ) be a purely non-atomic measure space, and let 1 < p < ∞. If L(Ω,Σ, μ) is isomorphic, as a Banach space, to L(Ω,Σ, μ) for some purely atomic measure space (Ω,Σ, μ), then there is a measurable partition Ω = Ω1 ∪Ω2 such that (Ω1,Σ ∩ Ω1, μ|Σ∩Ω1) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ ⊆ Ω2. In particular, L(Ω,Σ, μ) is isomorphic to l.
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ژورنال
عنوان ژورنال: Mathematische Annalen
سال: 1973
ISSN: 0025-5831,1432-1807
DOI: 10.1007/bf01428265