The subsequence of a sequence A selected by a selection rule r is that with r(A|n − 1) = yes. The sequence of selected places are those ni such that r(A|ni − 1) = yes. Then for a given selection rule r and a given real A, we generate a sequence n0, n1 , . . . of selected places, and we say that a real is stochastic with respect to admissible selection rules iff for any such selection rule, either the sequence of selected places is finite, or
There is a canonical way of turning a selection rule into a martingale. Namely, initially have F0(λ) = 1. Then for n0, the first selected place, double the bet on the 1’s. That is, for example, if n0 = 2 so that 2 is the first selected place, we would have F1(00) = F(10) = 1, F1(σ) = F1(τ) = 2, ∀ σ with 01 ≤ σ, and τ with 11 ≤ τ, and we’d raise F1(0) = F1(1) = 1 + 1, 2 and F1(λ) = 1 + 1/2, and we would continue this process in the obvious way. Clearly this process will give a martingale with limsFs(λ). The martingale is computable if the selection rule is.