{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mid1 - Write your name clearly on all sheets of paper you...

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

View Full Document Right Arrow Icon
Write your name clearly on all sheets of paper you will turn in and possibly number them. Write clearly and large enough to be easily readable. Your poofs must be complete and clearly written. Each question is worth the number of points indicated. Fifty five points will grant you an A but you have to solve at least one between ex. 1 and ex. 2. 1 (15 pts) Let f : R R 2 be a C 1 function ( i.e. f is continuous, differentiable and the derivative is continuous). Prove that μ ( f ( R )) = 0, where μ is the Lebesgue measure on R 2 . 2 Let M 1 and M 2 be two σ -algebras of subsets of X . Let μ 1 be a measure defined on M 1 and μ 2 be a measure on M 2 . a) (10 pts) Give a definition for the measure μ = μ 1 + μ 2 . On which σ -algebra M is μ defined? Show that your definition defines a measure. b) (10 pts) Show that if μ 1 and μ 2 are complete than μ = μ 1 + μ 2 is complete. c) (10 pts) Show that if μ 1 M and μ 2 M are σ -finite than μ = μ 1 + μ 2 is also σ -finite.
Image of page 1

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

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

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern