Order Completeness of L 1 with Applications to Stochastics

D. Plachky
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D. Plachky: Institute of Math. Statistics

A chapter in Approximation, Probability, and Related Fields, 1994, pp 395-399 from Springer

Abstract: Abstract A short and straightforward proof of the order completeness of L 1(Ω, Α, μ) for arbitrary positive measure spaces (Ω, Α, μ) is given including the fact that the corresponding least upper bound coincides with the least upper bound of a countable subset. As an application a characterization of atomless probability measures is rederived, a refinement of the Halmos-Savage result concerning families of probability measures dominated by a σ-finite measure is treated, some basic properties concerning least upper bounds of bounded, finitely additive set functions are presented, and a Riesz type decomposition for finitely additive measures, which includes the Hammer-Sobczyk and the Hewitt-Yosida decomposition, is proved.

Date: 1994
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4615-2494-6_30

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DOI: 10.1007/978-1-4615-2494-6_30

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