1.3. MEASURES ON PRODUCT SPACES. 49 Theorem 1.3.4 LetX and Y be ra...

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1.3. MEASURES ON PRODUCT SPACES. 49 Theorem 1.3.4 LetX and Y be randomelements defined ona commonproba-bility space. (a) If X and Y are independent, then so are g(X) and h(Y ) where g and h are appropriately measurable functions. (b) If g and h in part (a) are real valued, then E[g(X)h(Y )] = E[g(X)]E[h(Y )] provided all three functions are integrable. Proof. Part (a) is left as Exercise 1.3.6. For (b), let P = Law[g(X)] and Q = Law[h(Y )]. By part (a), Law[(g(X), h(Y ))] = P × Q. By the law of the 52 CHAPTER 1. MEASURE SPACES. 1.3.2 Show that the product of counting measures on two discrete sets is count-ing measure on the product set. 1.3.3 Let f : IN × IN −→ IR be given by f(n, m) = 1 if n = m, −1 if n = m − 1, 0 otherwise. Show that (∞ n=1 $ (∞ m=1 f(n, m) & ̸= (∞ n=1 $ (∞ m=1 f(n, m) & . Is f integrable w.r.t. # × #? Determine the answer to this last question directly from the definition of integrability. 1.3.4 Show that (1.46) and (1.47) are equivalent. 1.3.5 Let (X1, X2, X3) have distribution on IR3 which puts measure 1/4 on each of the points (0, 0, 0), (1, 1, 0), (1, 0, 1), and (0, 1, 1). (a) Show that the distribution of any pair of the r.v.’s is the product of two copies of the measure on IR which puts measure 1/2 at each point 0 and 1. Conclude that (X1, X2, X3) are pairwise independent. (b) Show that (X1, X2, X3) are not jointly independent. 1.3.6 Prove Theorem 1.3.4 (a). 1.3.7 Suppose (1.48) holds for all bounded real valued functions g and h on the ranges of X and Y , respectively. Show X and Y are independent. 1.3.8 Show that the product of two σ-finite measure spaces is also σ-finite. Hint: Let (Ω2, F1, µ1) and (Ω2, F2, µ2) be σ-finite, say Ω1 = !∞ n=1 An where µ1(An) < ∞, and Ω2 = !∞ m=1 Bm where µ2(Bm) < ∞. Let Cnm = An ×Bm. The collection {Cnm : n, m = 1, 2,...} is countable by Exercise 1.1.6 (a). 1.3.9 Define by induction the product of n measurable spaces and the product of n measure spaces. 1.3. MEASURES ON PRODUCT SPACES. 53 1.3.10 Evaluate the integral , f d(µ1 × µ2) when f, µ1, and µ2 are as given below. (a) µ1 and µ2 are both m, Lebesgue measure, and f(x1, x2) = 1 if x2 1 + x2 2 ≤ 1, 0 otherwise. (b) µ1 and µ2 are the same as in (a) but f(x1, x2) = e−x1/x2 if 0 ≤ x1 < ∞ and 0 < x2 < 1, 0 otherwise. (c) µ1 and µ2 are counting measure on IN, and f(n, m)=2−(n+m) . (d) µ1 is Lebesgue measure, µ2 is counting measure on IN, and f(x, n) = xn if 0 < x1 ≤ 1/2, 0 otherwise. (e) µ1 is Lebesgue measure m2 on IR2, µ2 is counting measure on {0, π, 2π}, and f((x1, x2), x3) = exp[|x1| − |x2| + cos(x3)]. 1.3.11 Generalize Theorem 1.3.4 to the case of n random elements. 1.3.12 Suppose X1, X2, ..., Xn are independent r.v.’s with c.d.f.’s given by F1, F2, ..., Fn, respectively. What is the c.d.f. of Y = max { X1, X2, ..., Xn}? 1.3.13 Let f = (f1, f2): (Ω, F) −→ (Λ1, G1) × (Λ2, G2). Show that each of the component functions f1 and f2 are measurable, i.e. that f1 : (Ω, F) −→ (Λ1, G1). 1.3.14 Prove or disprove the following: Let f : Ω1 × Ω2 −→ Λ where (Ωi, Fi), i = 1, 2, and (Λ, G) are measurable spaces. Suppose f(ω1, ω2) is a measurable function of ω1 for each fixed ω2 and is also a measurable function of ω2 for each fixed ω1. Then f is measurable (Ω1, F1) × (Ω2, F2) −→ (Λ, G). You may assume the existence of non–Borel sets V ⊂ IR. (Hint: Let V be such a subset of IR and consider the indicator of D = {(x1, x2) ∈ IR2 : x1 = x2 ∈ V }.) Contrast this claim with Theorem 1.3.5. 1.3.15 Give an alternate solution to Exercise 1.2.37 using Fubini’s theorem. State and prove a possible result when the fi’s are not required to be nonnegative. 1.3.16 Let µ1 and µ2 be σ-finite measures on (Ω, F), and let ν be a σ-finite measure on (Λ, G). (a) Show µ1 + µ2 is σ-finite. See Exercise 1.2.35 for the definition of µ1 + µ2. Hint: Similarly to Exercise 1.3.8, but consider Cnm = An ∩ Bm. (b) Show that (µ1 + µ2) × ν = (µ1 × ν)+(µ2 × ν). 54 CHAPTER 1. MEASURE SPACES. 1.3.17 Let A be a measurable set on the product space (Ω2, F1, µ1)×(Ω2, F2, µ2) and for each ω1 ∈ Ω1 put A(ω1) = { ω2 : (ω1, ω2) ∈ A } . (Note: Drawing a picture of an example when Ωi = IR may be helpful here.) Show that µ2(A(ω1)) = 0 µ1-a.e. iff (µ1 × µ2)(A) = 0. Apply this to determine m2({(x1, x2) : x1 = x2 }). 1.3.18 Find mn({ (x1, x2, ..., xn) : xi = xj for some i ̸= j}). (Hint: consider the special cases n = 2 and n = 3 first. In these cases, you can “see” the set.) 1.3.19 Determine whether or not the following function is integrable w.r.t. m2: f(x1, x2) = (x1x2)/(x2 1 + x2 2)2 if |x1| ≤ 1 and |x2| ≤ 1, 0 otherwise.

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