By H. Jerome Keisler
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Additional resources for An Infinitesimal Approach to Stochastic Analysis
I=1 Ωi , i=1 Ai ) = (ΩC , AC ) is the product measurable set of (Ωi , Ai ) (1 ≤ i ≤ n). The order on ΩC is defined similarly to that of Section 2. Throughout this section, we assume ϕ to be an increasing measurable function from (ΩC , AC ) to (S, S), where S is the power set of S and the definition of an increasing function is similar to that of Section 2. Furthermore, we assume that Ωi (1 ≤ i ≤ n) and S are endowed with discrete topology. Then, ϕ is a continuous function. Generally a subset W of an ordered set Ω is called increasing iff x ∈ W and x ≤ y imply y ∈ W .
5 49 Age and Shock Number Model The system is replaced at age T , shock number N or system failure, whichever occurs first.
I) Suppose that W is an increasing subset of Ω1 × Ω2 . Then we have W = ∪m j=1 (Aj × Bj ), where Aj (1 ≤ j ≤ m) are nonempty subsets of Ω1 such that A1 ⊂ · · · ⊂ Am and Ai = Aj (i = j) hold, and Bj (1 ≤ j ≤ m) are nonempty subsets of Ω2 such that ∪m k=j Bk (1 ≤ j ≤ m) are increasing subsets of Ω2 . Then, W = (PΩ1 W ) × (PΩ2 W ) holds iff m = 1 holds. n (ii) Suppose that W is an increasing subset of i=1 Ωi . Then, W = n i=1 (PΩi W ) holds iff W has the minimal element. 2. α α α (i) aα 0 + a1 − b1 > [a0 + a1 − b1 ] holds for 0 < α < 1, a0 ≥ a1 > b1 > 0.