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In a probability space (Ω,σ,P), for α ⊂ σ a sub-σ field, in general the best approximation in L by elements of L(α) has not a unique solution. For the election between these, we prove the convergence P-almost surely of the conditional r-means, when r → ∞, to one solution, which we call conditional mid-range. This is characterized for each ω ∈ Ω by the mid-range, of one regular conditional distribution Q(ω, ·).
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