Corollary 5 71 sections of measurable functions are

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Corollary 5. 7.1 Sections of measurable functions are measurable. That is, if then Proof. Since X is 81 x 82/ S measurable, we have for A e S that and hence by the previous results However (X- 1 (A))w 1 ={w2 : X(wt, W2) E A} ={W2 : Xw 1 (w2) E A}= (Xw 1 )- 1 (A), which says Xw 1 is 82/S measurable. 0
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5.8 Probability Measures on Product Spaces 147 5.8 Probability Measures on Product Spaces We now consider how to construct probability measures on the product space (Qt x !lz, Bt x Bz). In particular, we will see how to construct independent random variables. Transition Functions. Call a function a transition function if (i) for each Wt, K(wt. ·)is a probability measure on Bz, and (ii) for each Az e Bz, K(·, Az) is Bt/8([0, 1]) measurable. Transition functions are used to define discrete time Markov processes where K (cut. Az) represents the conditional probability that, starting from Wt, the next movement of the system results in a state in Az. Here our interest in transi- tion functions comes from the connection with measures on product spaces and Fubini's theorem. Theorem 5.8.1 Let Pt be a probability measure on Bt, and suppose K : !lt x Bz [0, 1] is a transition function. Then K and Pt. uniquely determine a probability on Bt x Bz via the formula P(At x Az) = { K (cut. Az)Pt (dwt), (5.20) }AI fora// At x Az E RECT. The measure P given in (5.20) is specified on the semialgebra RECT and we need to verify that the conditions of the Combo Extension Theorem 2.4.3 on page 48 are applicable so that P can be extended to a(RECT) = Bt x Bz. We verify that Pis a-additive on RECT and apply the Combo Extension The- orem 2.4.3. Let { A(n) x A(n) n > 1} t 2 ' - be disjoint elements of RECT whose union is in RECT. We show 00 00 P(L X = L X n=t n=t Note if Ln x =At x Az, then 1A 1 (wt)1A 2 (wz) = \ 1A 1 xA2(Wt. wz) = L 1 wz) n = L1Atn)(Wt)lA<nJ(lUz). n 1 2
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148 5. Integration and Expectation Now making use of the series form of the monotone convergence theorem we have the following string of equalities: P(At x A2) = { 1A 1 (wt)K(wt, A2)P1(dwt) ln1 = { [ { 1A 1 (WI)lA 2 (W2)K(wi,dW2)]Pt(dwt) ln1 ln2 = { [ { L)A<n>(wt)lA<n>(W2)K(wt,dW2)]Pt(dwt) lnl lnz n I 2 = { L) { lA<nJ(WI)lA<n>(W2)K(wt.dW2)]Pt(dwt) n I 2 = L { lA(nJ(Wt)[ { 1A<nJ(W2)K(Wt,dW2)]Pt(dwt) n lnl I ln2 2 = L { n lnl I = L 1 K (WI, An))Pt (dwt) n A\n) = L X Ain)). n 0 Special case. Suppose for some probability measure P2 on 82 that K (wt. A2) = P 2 (A 2) . Then the probability measure P, defined in the previous result on Bt x Bz is P(At x A2) = Pt(At)Pz(Az) . We denote this P by Pt x Pz and call P product measure . Define u-fields in nt x n2 by B'f = {At x nz :At e 8t} = {Qt X A2 : A2 E 82} . With respect to the product measure P = Pt x P2, we have l3lj JL since P(At x n2 n n1 x A2) = P(At x A2) = P1 (At)P2(A2) = P(At x n2)P(nt x A2). Suppose X ; : (Q;, 8;) (IR, 8(1R)) is a random variable on n; fori= 1, 2. Define on S'2t X n2 the new functions
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5.9 Fubini's theorem 149 With respect toP = Pt x P2, the variables Xf and are independent since P[Xf::; x, y] = Pt x P2({(w1, w2): Xt (wt) ::; x, X2(W2) :=: y}) = Pt x P2({w1: Xt(wt)::::: x} x {W2: X2(W2)::::: y}) = Pt({Xt(wt)::::: x})P2({W2: X2(W2)::::: y}) = P({(wt. W2): Xf(wt. W2)::::: x}) P({(wt, W2): W2) ::; y}) = P[Xf ::: X ::: y ].
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