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H/ We consider the probalility space (CO, 1 1] B, r) J wher...

H/ We consider the probalility space (CO, 1 1] B, r) J where B is the Borel 6-algebra of [0,1], M is the Lebesgue measure. Given two random variables: f, 0x1= X2 f2 (K) = 1-X2 aj Compute their distribution functions and whether they are identically con clude distributed or not. b) Are they ...

Assignment 2 Let J = { [-: - S, t) I S,t > o} a, Sh...

Assignment 2 Let J = { [-: - S, t) I S,t > o} a, Show that o(j) BBR). ( B(R) = Borel Sigma argebral b, show that each of following sets belongs to the 5-algebra ocj) : [=5,-1) , [3.8) , (-5,8] Assignment 3 Let ( X, A,N) be a measure space, where 2 is a probability Measure. Let An, A2...

(8 points) 1. Let /i be continuous functions on the interval...

(8 points) 1. Let /i be continuous functions on the interval [0.1] and suppose that si- / uniformly on the interval. Then, using Lebesgue mensure " and the Lebesgue integral, prove that (8 points) 2. Are the simple functions dense in 1°0(IR) (in the 100 topology)? Why or why not? (...

(1) Suppose that f1 : [0,1] R and f2 : (1, 2] R are measurab...

(1) Suppose that f1 : [0,1] R and f2 : (1, 2] R are measurable functions. Define X € = X € Show that F is a measurable function on [0,2]. (2) Let E C R have the property that m. (EnK) = 0 for all compact subsets K CR. Prove that m. (E) = 0. (3) Given an integrable function f on R, p...

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