HW1-320-Solutions

HW1-320-Solutions - Math 320 Measure Theory and Integration...

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Math 320 Measure Theory and Integration Assignment 1: & -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0 ; 1] which admit decimal expansions such that the digits 2 ; 4 ; 6 ; 8 all appear at least once. (Thus, 0.0032004068 is an example while 0.232046 is not. For 0.9999. ..., write 1 instead.) Show that A is a Borel set. Solution. Let E i stand for the set of points in [0 ; 1] such that the digit 1 < i < 9 appears in the decimal expansions of the points. Let F ( i ) j denote the set of numbers with i appearing in the j -th decimal place. We have F ( i ) j = [ 10 j 1 k =0 [ k 10 j + i 10 j ; k 10 j + i +1 10 j ) . Then E i = [ 1 j =1 F ( i ) j . Note that E i is a Borel set. Finally, the set we want to describe is \ i =2 ; 4 ; 6 ; 8 E i with each E i a Borel set. Hence, it is Borel. 2. Given sets X and Y . Let f : X ! Y be any function into Y . (a) Show that if A is a & -algebra of subsets of Y , then f 1 ( A ) is a & -algebra of subsets of X . (b) Show that ( f 1 ( S )) = f 1 ( ( S )) , for any collection S of subsets of Y . Here ( K ) represents the & -algebra generated by K . Solution. (a) [1] X 2 f 1 ( A ) since Y 2 A and f ( X ) = Y . [2] Note we have f 1 ( A c ) = ( f 1 ( A )) c for any A . (In general it is not true that f ( A c ) = f ( A ) c .) We have A 2 f 1 ( A ) () A = f 1 ( B ) for some B 2 A : Thus A c = ( f 1 ( B )) c = f 1 ( B c ) 2 f 1 ( A ) (since B 2 A ). [3] Let A 1 ;A 2 ;::: 2 f 1 ( A ) . We want to show [ A i 2 f 1 ( A ) . We have A i = f 1 ( B i ) for i = 1 ; 2 ; 3 ::: with B i 2 A . Now, [ A i = [ f 1 ( B i ) = f 1 ( [ B i ) 2 f 1 ( A ) , since [ B i 2 A . (b)
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HW1-320-Solutions - Math 320 Measure Theory and Integration...

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