HW1-320-Solutions

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

This preview shows pages 1–2. Sign up to view the full content.

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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online