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Abstract. Let L be a linear space of real bounded random variables on the probability space (Ω, A, P0). There is a finitely additive probability P on A, such that P ∼ P0 and EP (X) = 0 for all X ∈ L, if and only if c EQ(X) ≤ ess sup(−X), X ∈ L, for some constant c> 0 and (countably additive) probability Q on A such that Q ∼ P0. A necessary condition for such a P to exist is L − L + ∞ ∩ L + ∞ = {0}, where the closure is in the norm-topology. If P0 is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on A, such that P ≪ P0 and EP (X) = 0 for all X ∈ L, if and only if ess sup(X) ≥ 0 for all X ∈ L. 1

Year: 2012

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