{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math630-3

# Math630-3 - Real Analysis Math 630 Homework Set#3 Chapter 3...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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.

Unformatted text preview: Real Analysis - Math 630 Homework Set #3 - Chapter 3 by Bobby Rohde 9-21-00 Problem 18 & Show that (v) does not imply (iv) in Propostion 18 by constructing a function f such that { x : f ( x ) > 0} = E , a given non-measurable set, and such that f assumes each value at most once. & Counter-Example Let E = the standard non-measurable set on [0, 1] Let f ( x ) = & x , x & E ¡ x , x ¢ E defined on domain [0, £ ) Thus { x : f ( x ) > 0} = E , and f ( x ) = ¤ has at most one solution for any ¤ , namely ¤ or - ¤ . Hence { x : f ( x ) = ¤ } is measurable ¥ ¤ but { x : f ( x ) > 0} is not measurable. Problem 19 & Let D be a dense set of real numbers. Let f be an extended real-valued function on & such that { x : f ( x ) > & } is measurable ¡ & ¢ D . Show that f is measurable. & Proof We wish to show that { x : f ( x ) > & } is measurable ¡ & ¢ D is equivalent to the condition { x : f ( x ) > & } is measurable ¡ & . Take £ ¤ D then define A n = { x : f ( x ) > ¥ n , with ¥ n ¢ ( £- 1 ¦¦¦¦¦ n , £ ) & D }. We know that ( £- 1 ¦¦¦¦¦ n , £ ) & D is non-empty ¡ n since, D is dense. Consider & n § 1 ¨ A n © & n § 1 ¨ ¡ x : f ¢ x £ ª £ « 1 ¦¦¦¦¦ n ¤ = { x : f ( x ) ¬ £ }. But { x : f ( x ) ¬ £ } © & n § 1 ¨ A n since ¡ n , ¥ n < £ . Thus & n § 1 ¨ A n = { x : f ( x ) ¬ £ } which means that { x : f ( x ) ¬ £ } is measurable and hence by Proposition 18, { x : f ( x ) > £ } is measurable, so { x : f ( x ) > & } is measurable ¡ & . QED Problem 20 & Show that the sum and product of two simple functions are simple. & Proof £ ¤ A & B ¥ ¤ A ¦ ¤ B ­ A & B = ¡ 1, if x ¢ A & B 0, if x ¤ A & B ­ A ® ­ B = ¡ 1, if x ¢ A and x ¢ B 0, if x ¤ A or x ¤ B But x ¢ A and x ¢ B ¯ x ¢ A & B and, x ¤ A or x ¤ B ¯ x ¤ A & B, thus ­ A & B § ­ A ® ­ B , QED. MATH630-3.nb 2 & ¡ A & B ¢ ¡ A £ ¡ B ¤ ¡ A ¥ ¡ B & A & B = ¡ 1, if x ¡ A & B 0, if x ¢ A & B & A £ & B ¤ & A ¥ & B = & A £ & B ¤ & A ¢ B = ¡ 1, if x ¡ A or x ¡ B 0, if x ¢ A and x ¢ B But x ¡ A or x ¡ B ¦ x ¡ A & B and, x ¢ A and x ¢ B ¦ x ¢ A & B, thus & A & B § & A £ & B ¤ & A ¥ & B . QED....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern