# hw - 205A Homework#1 due Monday 13 September 1[Bill 2.4 Let...

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205A Homework #1 , due Monday 13 September. 1. [Bill. 2.4] Let F n be classes of subsets of S . Suppose each F n is a ﬁeld, and F n ⊂ F n +1 for n = 1 , 2 ,... . Deﬁne F = n =1 F n . Show that F is a ﬁeld. Give an example to show that F need not be a σ -ﬁeld. 2. [Bill. 2.5(b)] Given a non-empty collection A of sets, we deﬁned F ( A ) as the intersection of all ﬁelds containing A . Show that F ( A ) is the class of sets of the form m i =1 n i j =1 A ij , where for each i and j either A i,j ∈ A or A c ij ∈ A , and where the m sets n i j =1 A ij , 1 i m are disjoint. 3. [Bill. 2.8] Suppose B σ ( A ), for some collection A of subsets. Show there exists a countable subcollection A B of A such that B σ ( A B ). 4. Show that the Borel σ -ﬁeld on R d is the smallest σ -ﬁeld that makes all continuous functions f : R d R measurable. 5. [Durr. 1.2.5 or 1.2.4] A function f : R d R is lower semicontinuous (l.s.c.) if liminf y x f ( y ) f ( x ) for all x. A function is upper semicontinuous (u.s.c.) if limsup y x f ( y ) f ( x ) for all x. Show that, if f is l.s.c. or u.s.c., then f is measurable. 1

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205A Homework #2 , due Monday 20 September. 1. [similar Bill. 2.15] Let B be the Borel subsets of R . For B ∈ B deﬁne μ ( B ) = 1 if (0 ) B for some ε > 0 = 0 if not (a) Show that μ is not ﬁnitely additive on B . (b) Show that μ is ﬁnitely additive but not countably additive on the ﬁeld B 0 of ﬁnite disjoint unions of intervals ( a,b ]. 2. Show that, in the deﬁnition of “a probability measure μ on a measurable space ( S, S )”, we may replace “countably additive” by “ﬁnitely additive, and satisﬁes if A n φ then μ ( A n ) 0 . 3. [similar Durr. A.2.1] Give an example of a measurable space ( S, S ), a collection A and probability measures μ and ν such that (i) μ ( A ) = ν ( A ) for all A ∈ A (ii) S = σ ( A ) (iii) μ 6 = ν . Note: this can be done with S = { 1 , 2 , 3 , 4 } 4. [similar Durr. A.3.1] Let μ be a probability measure on ( S, S ), where S = σ ( F ) for a ﬁeld F . Show that for each B ∈ S and ε > 0 there exists A ∈ F such that μ ( B Δ A ) < ε . 5. [similar Durr. A.4.3] Let g : [0 , 1] R be integrable w.r.t. Lebesgue measure. Let ε > 0. Show that there exists a continuous function f : [0 , 1] R such that R | f ( x ) - g ( x ) | dx ε . 2
205A Homework #3 , due Monday 27 September. 1. Use the monotone convergence theorem to prove the following. (i) If X n 0, X n X a.s. and EX n < for some n then EX n EX . (ii) If E | X | < then E | X | 1 ( | X | >n ) 0 as n → ∞ . (iii) If E | X 1 | < and X n X a.s. then either EX n EX < or else EX n ↑ ∞ and E | X | = . (iv) If

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## This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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hw - 205A Homework#1 due Monday 13 September 1[Bill 2.4 Let...

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