No3(2004) - MATH 587 Assignment 3 Due November 18, 2004 (1)...

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MATH 587 Assignment 3 Due November 18, 2004 (1) Let µ be a measure, and λ 1 and λ 2 be signed measures on (Ω , F ). Prove that (a) if λ 1 µ and λ 2 µ , then ( λ 1 + λ 2 ) µ . (b) λ 1 ¿ µ ⇐⇒ | λ 1 |¿ µ ⇐⇒ λ + 1 1 ¿ µ . (c) if λ 1 ¿ µ and λ 2 µ , then λ 1 λ 2 . (d) if λ 1 ¿ µ and λ 1 µ , then λ 1 =0. Note: if λ 1 2 are signed measures, we deFne λ 1 λ 2 to mean | λ 1 |⊥| λ 2 | . (2) Let X be a non-negative r.v. on a probability space (Ω , F ,P ), and let 1 p< . Use ±ubini’s theorem to show that EX p = R 0 px p 1 P { X>x } dx . (3) Prove that for any distribution function F ( x )= P { X x } and any a 0, we have Z −∞ F ( x + a ) F ( x ) dx = a. (4) Let X n ,n 1 be a sequence of Fnite-valued random variables on the probability space (Ω , F ,P ). Show that it is always possible to Fnd a sequence a n ,n 1 of constants such that X n a n 0 a.s. [Hint: Consider n =1 P ( | X n a n | > 1 n ) for suitable a n .] (5) Suppose that { X n ,n 1 } is a sequence of non-negative r.v.’s on a probability space (Ω
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