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practice_sol - Stat 310A/Math 230A Theory of Probability...

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Stat 310A/Math 230A Theory of Probability Practice Midterm Solutions Andrea Montanari October 28, 2010 Problem 1 Consider the measurable space (Ω , F ), with: Ω = { 0 , 1 } N the set of (infinite) binary sequences ω = ( ω 1 , ω 2 , ω 3 , . . . ); F the σ -algebra generated by cylindrical sets (equivalently the σ -algebra generated by sets of the type A i,x = { ω : ω i = x } for i N and x ∈ { 0 , 1 } ). (a) Let f : Ω [0 , 1] be defined by f ( ω ) X n =1 ω n 2 n . (1) for ω = ( ω 1 , ω 2 , . . . ). Is f measurable? Prove your answer. (As usual, assume [0 , 1] to be endowed with the Borel σ -algebra.) Solution : Yes, f is measurable. Indeed consider the following collection of intervals P = { B k,n = [ k/ 2 n , ( k + 1) / 2 n ) : , n, k N , n 0 , 0 k 2 n - 1 } ∪ n { 1 } o . (2) This is a π -system (indeed if n m , B k,n B l,m is either empty or equal to B k,n ). Further σ ( P ) = B [0 , 1) . Indeed any interval of the form [ k/ 2 n , l/ 2 n ) is disjoint union of a finite colection of intervals in P . As a consequence any interval ( a, b ), 0 a < b 1 is in σ ( P ) since we can find monotone sequences a n = k n / 2 n > a , b n = l n / 2 n < b with a n a , b n b , whence ( a, b ) = n n 0 [ a n , b n ). By taking union intevals in P , we conclude that all intervals of the form [0 , b ), ( a, 1] are in σ ( P ) as well. This completes the proof that σ ( P ) = B [0 , 1) . It is therefore sufficient to show that f - 1 ( B k,n ) ∈ F for k, n as above. If the binary expansion of k is k = k 1 2 n - 1 + k 2 2 n - 2 + · · · + k n , it is elementary to see that f - 1 ( B k,n ) = n ω such that ω 1 = k 1 , . . . , ω n = k n o \ n ω = ( k 1 , k 2 , . . . , k n , 1 , 1 , 1 , 1 , . . . ) o , (3) which is the intersection of two measuraable sets. (b) Let g : [0 , 1] Ω be defined by g ( x ) ( g 1 ( x ) , g 2 ( x ) , g 3 ( x ) , . . . ) , (4) g i ( x ) bb 2 i x cc mod 2 , (5) where bb a cc ≡ max { n N : n < a } . Is g a measurable mapping? Prove your answer. (Assume [0 , 1] and Ω to be endowed the same σ -algebras as above.) Solution : Consider the collection of sets P = { C n ( u n 1 ) : n N , u n 1 ∈ { 0 , 1 } n } ⊆ F defined by C n ( u n 1 ) = { ω : ω 1 = u 1 , ω 2 = u 2 , . . . , ω n = u n } . (6) 1
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It is clear that P is a π -system (indeed, for n m C n ( u n 1 ) C m ( v m 1 ) is either empty or coincides with C n ( u n 1 )) with σ ( P ) = F (indeed the events A i,x can be constructed as union of finitely many events C n ( u n 1 )). We have g - 1 ( C n ( u n 1 )) = x [0 , 1] : bb 2 x cc = u 1 mod 2 \ · · · \ x [0 , 1] : bb 2 n x cc = u n mod 2 . (7) It is therefore sufficient to show that any set of the form { x [0 , 1] : , bb 2 k x cc = u k mod 2 is Borel. Such a set is the union of a finite number of sets of the form J k,j = { x [0 , 1] : , bb 2 k x cc = j , with j N . But bb 2 k x cc = j if and only if 2 k x ( j, j + 1], and therefore J k,j are just intervals.
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