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# hw3 - Prove that i± f is monotone increasing then f is...

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Math 280A, Fall 2011 Homework 3 1. Problem 14 (page 116) 2. Problem 20 (page 117) 3. Let ( E i , E i ), i = 1 , 2 be measurable spaces. For a set A E 1 × E 2 and a point x E 1 we defne the “vertical cross section” A x := { y E 2 : ( x, y ) A } . The goal o± this problem is to show that A x ∈ E 2 whenever A ∈ E 1 ⊗ E 2 . (The analo- gous assertion ±or horizontal cross sections A y := { x E 1 : ( x, y ) A } is le±t to your imagination.) (a) Show that, ±or B 1 E 1 and B 2 E 2 , ( B 1 × B 2 ) x = b B 2 , x B 1 , , x / B 1 . (b) Fix x E 1 and defne G := { A ∈ E 1 ⊗ E 2 : A x ∈ E 2 } . Argue that (i) G contains the π -system C := { B 1 × B 2 : B i ∈ E i ±or i = 1 , 2 } (part (a) will be use±ul here), and (ii) that G is a σ -algebra (o± subsets o± E 1 × E 2 ). Deduce that G = E 1 ⊗ E 2 . 4. A ±unction f : R R is monotone increasing i± f ( x ) f ( y ) whenever x < y
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Unformatted text preview: . Prove that i± f is monotone increasing, then f is Borel measurable, in the sense that f-1 ( B ) ∈ R ±or each B ∈ R . (Recall that R denotes the Borel subsets o± R .) [Hint: Examine the inverse images f-1 (-∞ , x ] o± hal±-lines.) 5. Let X and Y be random variables defned on some probability space (Ω , F , P ), such that P [( X, Y ) = (0 , 1)] = P [( X, Y ) = (0 ,-1)] = P [( X, Y ) = (1 , 0)] = P [( X, Y ) = (-1 , 0)] = 1 / 4 . (a) Compute E [ X ], E [ Y ], and E [ XY ], thereby confrming that E [ XY ] = E [ X ] · E [ Y ]. (b) Are X and Y independent?...
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