real-analysis-hw3-with-solution-2007-9-25

real-analysis-hw3-with-solution-2007-9-25 - Real Analysis...

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

View Full Document Right Arrow Icon
Real Analysis Homework 3, due 2007-10-3 in class Show Your Work to Each Problem 1. (20 points) Let f : R n R be a continuous function. De f ne the collection of sets P as X = © B R : f 1 ( B )i sm ea su rab l e ª . Does P form a σ -algebra? If B R is a Borel set, does it follow that f 1 ( B )isaBore l set? Give your reasons. Solution: Clearly , R P . If E P , by f 1 ( E c )= ¡ f 1 ( E ) ¢ c (0.1) we know that E c P also. Similarly if { E k } k =1 is a collection of subsets of P ,thenby f 1 Ã [ k E k ! = [ k f 1 ( E k )a n d f 1 Ã \ k E α ! = \ k f 1 ( E α ) (0.2) we know that k E k P and k E α P . Hence P is a σ -algebra. Similarly we can show that the set Λ = © B R : f 1 ( B ) is Borel measurable ª forms a σ -algebra. For any open set O R , the set f 1 ( O ) is also open. Hence f 1 ( O ) Λ . The sameforclosedset.Henceallopensetsandclosedsetsarecontainedin Λ . Since Λ is a σ -algebra, if B R is a Borel set, then f 1 ( B )musta lsobeaBore lset . ¤ 2. (20 points) Do Exercise 12 in p. 48. Hint: You can use Theorem 3.29. Solution: We f rst show that E 1 × E 2 R 2 is measurable.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/05/2009 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

Page1 / 3

real-analysis-hw3-with-solution-2007-9-25 - Real Analysis...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online