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Additional resources for Accuracy of MSI testing in predicting germline mutations of MSH2 and MLH1 a case study in Bayesian m
Ii) If also Bi ⊆ i , i = 1, 2, . . , n, and F = B1 × · · · × Bn , then show that E ∩ F = (A1 × · · · × An ) ∩ (B1 × · · · × Bn ) = (A1 ∩ B1 ) × · · · × (An ∩ Bn ). 17. For i = 1, 2, . . , n, let Ai , Bi , Ci ⊆ i and set E = A1 × · · · , ×An , F = B1 × · · · × Bn , G = C1 × · · · × Cn . Suppose that E, F, and G are all = and that E = F + G. Then show that there exists a j with 1 ≤ j ≤ n such that A j = B j + C j while Ai = Bi = Ci for all i = j. 18. In reference to Theorem 7, show that C is still a field, if Ai is replaced by a field Fi , i = 1, 2.
If we have n ≥ 2 measurable spaces ( i , Ai ), i = 1, . . , n, the product measurable space ( 1 ×· · ·× n , A1 ×· · ·×An ) is defined in an analogous way. In particular, if 1 = · · · = n = and A1 = · · · = An = B, then the product space ( n , B n ) is the n-dimensional Borel space, where n = × · · · × , B n = B × · · · × B (n factors), and B n is called the n-dimensional Borel σ -field. The members of B n are called the n-dimensional Borel sets. Now we consider the case of infinitely (countably or not) many measurable spaces ( t , At ), t ∈ T , where the (= ) index set T will usually be the real line or the positive half of it or the unit interval (0, 1) or [0,1].
N. , μ(A1 ) ≤ μ(A2 ), A1 , A2 ∈ A, A1 ⊆ A2 . , μ 2, . . ∞ j=1 Aj ≤ ∞ j=1 μ(A j ), A j ∈ A, j = 1, Proof. ∞ (i) We have nj=1 A j = , j=1 B j , where B j = A j , j = 1, . . , n, B j = j = n + 1, . . ∞ n Then μ( nj=1 A j ) = μ( ∞ j=1 B j ) = j=1 μ(B j ) = j=1 μ(B j ) = n μ(A ). j j=1 (ii) A1 ⊆ A2 implies A2 = A1 +(A2 − A1 ), so that μ(A2 ) = μ[A1 +(A2 − A1 )] = μ(A1 ) + μ(A2 − A1 ) ≥ μ(A1 ). From this, it also follows that: A1 ⊆ A2 implies μ(A2 − A1 ) = μ(A2 )−μ(A1 ), provided μ(A1 ) is finite.