Chapter 7 Infinite product spaces

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12 CHAPTER 7. INFINITE PRODUCT SPACES Lemma 7.18 Let (M, r) be a compact metric space. Equip M N with the metric d(x, y) =Â r(xn, yn)2 n 1 n (7.29) whenever x =(xn) 2 M N and y =(yn) 2 M N . Then (M N , d) is a compact metric space. Moreover, if C1,...,Cn are compact sets in (M, r), then C1 ⇥···⇥Cn ⇥ M ⇥ M ⇥··· is compact in (M N , d). Proof. The first part is a reformulation of Tychonoff’s theorem for product spaces. The second part is checked directly. Now we are finally ready to prove Kolmogorov’s theorem: Proof of Theorem 7.16. Consider the set S = A1 ⇥···An ⇥ M ⇥···: Ai compact in M 8i Then, as is easy to check, S is a semialgebra. Let µ : S ! [0, 1] be defined by (7.30) µ(A1 ⇥···⇥An ⇥ M ⇥···)=µn(A1 ⇥···⇥An) (7.31) The consistency condition for µn shows that this gives the same number regardless of the representation of the set A1 ⇥···⇥An ⇥ M ⇥···. Let S be the set of all finite disjoint unions of elements from S . By Lemma 7.14, S is an algebra; we extend µ to a set function ¯µ : S ! [0, 1] by additivity. A similar argument to that used in the proof of Theorem 7.15 now shows that ¯µ is well-defined and finitely additive. It remains to show that ¯µ is countably additive. To that end, let A1, A2, ···2S be disjoint with A = S n 1 An 2 S . Define DN = S n>N An and note that DN 2 S as well. Then A1,...,An, Bn are disjoint and so finite additivity of ¯µ implies ¯µ(A) = ¯µ(A1)+···+ ¯µ(An)+ ¯µ(Dn) (7.32) We thus need to show that ¯µ(Dn) ! 0 as n ! •. Here we use the topology: By Lemma 7.17 it suffices to show 8e > 0 8B 2 S 9C 2 S : C ⇢ B compact & ¯µ(B \ C) < e. (7.33) But B 2 S is a finite disjoint union of Bi 2 S and so it clearly suffices to prove this for B 2 S . However, every B 2 S is compact by Lemma 7.18 and so we can take C = B above and the regularity property holds trivially. By Lemma 7.17, we have ¯µ(Dn) ! 0 and ¯µ is thus countably additive on S . The Carathéodory extension theorem now implies the existence of a unique extension of ¯µ to s(S )= B ⌦N . The proof in the textbook bundles Lemma 7.17 with the proof of Tychonoff’s theorem. We prefer to separate these to highlight the role of Borel regularity (Lemma 7.17) which is a property of separate interest. Note also that the consistency condition holds only for probability; extensions of other measures are not meaningful. In fact, it is known that one cannot construct an infinite product of Lebesgue measures on R.
7.5. TWO ZERO-ONE LAWS 13 7.5 Two zero-one laws We will demonstrate the usefulness of the infinite product spaces by proving two Zero-One Laws. We demonstrate these on an example of percolation. Example 7.19 Percolation : Let p 2 [0, 1] and let B denote the set of edges of Z d . (We regard the latter as a graph with vertices at points of R d with integer coordinates and edges between any two points with Euclidean distance one.) Let (h b) b2B be i.i.d. Bernoulli(p). We may view each h =(h b) as a subgraph of (Z d , B)—with edge set {b 2 B : h b = 1}. The question is: Does this graph have an infinite connected component? We define E• = {h : h has an infinite connected component}. (7.34) We refer to edges with h b = 1 as occupied and those with h b = 0 as vacant. To set the notation, we will denote by Pp the law of h with parameter p. First we have to check that E• is an event: Lemma 7.20 E• is measurable (w.r.t. the product s-algebra on {0, 1} B ). Proof. Let F denote the product s-algebra on {0, 1} B . We have to show E• 2 F . For each n 1 and each x 2 Z d , consider the event En(x) = x connected to (x +[ n, n] d ) c in h (7.35) that x is connected by a path of occupied edges to the boundary of a box of side 2n centered at x. Since En(x) depends only on a finite number of hb’s, it is measurable. But E• = \ [ En(x) (7.36) and so E• 2 F as well. n 1 x2Zd Next we observe that E• does not depend on the status on any given finite number of edges. Indeed, if an edge is occupied, making it vacant may increase the number of infinite component but it won’t destroy them while. Similarly, making a vacant edge occupied may connect two infinite components together but if there is none, it will not create one. In a more technical language, E• is a tail event according to the following definition: Definition 7.21 Let X1, X2,... be random variables. Then T = T n 1 s(Xn, Xn+1,...) is the tail s-algebra and events from T are called tail events. Here is our first zero-one law: Theorem 7.22 [Komogorov’s Zero-One Law] Let X1, X2,... be independent and let T = T n 1 s(Xn, Xn+1,...) be the tail s-algebra. Then 8A 2 T : P(A) 2{0, 1}. (7.37)
Page 1 and 2: Chapter 7 Infinite product spaces S
Page 3 and 4: 7.2. PROOF OF BOREL ISOMORPHISM THE
Page 5 and 6: 7.2. PROOF OF BOREL ISOMORPHISM THE
Page 7 and 8: 7.3. PRODUCT MEASURE SPACES 7 Thus
Page 9 and 10: 7.3. PRODUCT MEASURE SPACES 9 Theor
Page 11: 7.4. KOMOGOROV’S EXTENSION THEORE
Page 15 and 16: 7.5. TWO ZERO-ONE LAWS 15 Corollary
Page 17: 7.5. TWO ZERO-ONE LAWS 17 However P

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