Abstract
LetB be a Banach space with the Radon-Nikodym property and (S, ϕ, μ) a probability space. Then anf: S→B satisfies the strong law of large numbers if and only if there exists a Bochner integrable functionf 1 and a Pettis integrable functionf 2,f 2‖f 2‖=0 in the Glivenko-Cantelli norm, such thatf=f 1+f 2. The composition is unique.
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Dobric, V. The decomposition theorem for functions satisfying the law of large numbers. J Theor Probab 3, 489–496 (1990). https://doi.org/10.1007/BF01046091
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DOI: https://doi.org/10.1007/BF01046091