s3 - Math 8601 REAL ANALYSIS Fall 2010 Homework#3 Problems...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #3. Problems and Solutions. #1. Let f be a real function on R 1 . The image and the inverse image of a subset A ⊂ R 1 under f are correspondingly f ( A ) = { y : y = f ( x ) for some x ∈ A } , f- 1 ( A ) = { x : f ( x ) ∈ A } . Show that f ( f- 1 ( A )) ⊂ A ⊂ f- 1 ( f ( A )) for arbitrary A ⊂ R 1 . Give an example when f- 1 ( f ( A )) 6 = A . Solution. (i). y ∈ f ( f- 1 ( A )) = ⇒ y = f ( x ) for some x ∈ f- 1 ( A ) = ⇒ y = f ( x ) ∈ A . Therefore, f ( f- 1 ( A )) ⊂ A . (ii). x ∈ A = ⇒ y = f ( x ) ∈ f ( A ) = ⇒ x ∈ f- 1 ( f ( A )). Therefore, A ⊂ f- 1 ( f ( A )). (iii). For f ( x ) = sin x, A = { } , we have f- 1 ( f ( A )) = f- 1 (0) = { kπ : k = 0 , ± 1 , ± 2 , ···} 6 = A = { } . #2. Let f ( x ) be a continuous function on R 1 , and let B denote the Borel σ-algebra on R 1 . Show that M := f- 1 ( B ) := { E : E = f- 1 ( F ) for some F ∈ B} is a σ-algebra contained in B ....
View Full Document

{[ snackBarMessage ]}

Page1 / 2

s3 - Math 8601 REAL ANALYSIS Fall 2010 Homework#3 Problems...

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