Results 1 to 4 of 4

Math Help - Measure Theory Problem

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    16

    Measure Theory Problem

    Hi everyone
    I cannot find the solution to this problem,
    Let A a Lebesgue Measurable Set; Prove that given 0 \leq b \leq m(A) then exists B \subseteq A Lebesgue Measurable such that m(B) = b

    I appreciate your help
    Thanks
    everk.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by everk View Post
    Let A a Lebesgue Measurable Set; Prove that given 0 \leq b \leq m(A) then exists B \subseteq A Lebesgue Measurable such that m(B) = b
    Let {\mathstrut\chi}_A denote the characteristic function of A (so that {\chi\mathstrut}_A(t) =1 if t\in A, and {\mathstrut\chi}_A(t) = 0 if t\notin A). For each real number x, let \displaystyle f(x) = \int_{-\infty}^x\!\!{\mathstrut\chi}_A(t)\,dt = m(A\cap(-\infty,x]). Show that f is continuous and use the intermediate value theorem to deduce that there exists  c such that f(c)=b.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Does that work if A=(-\infty, 0)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Bruno J. View Post
    Does that work if A=(-\infty, 0)?
    It will certainly need a bit of adjustment if m(A) = \infty. I think that you would then have to start by looking at \displaystyle\int_x^X\!\!{\mathstrut\chi}_A(t)\,dt, which can be made arbitrarily large if  x is small enough and  X is large enough. Then reduce  X until the integral reaches the value  b.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. measure theory
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: July 5th 2011, 08:37 AM
  2. Measure Theory Problem
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: September 27th 2010, 08:16 PM
  3. Measure Theory
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 25th 2010, 07:20 AM
  4. Measure Theory problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 26th 2008, 01:55 PM
  5. measure theory
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: June 14th 2007, 01:47 AM

Search Tags


/mathhelpforum @mathhelpforum