Results 1 to 2 of 2

Math Help - Bounded, measurable?

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    53

    Bounded, measurable?

    1. Suppose A is a bounded set and that m^{*}(<br />
A\cap I)\leq am^{*}(I) for every interval I and 0 < a < 1. Prove that m^{*}(A)=0. What if A is unbounded?

    2. A is a subset of \mathbb{R} with the following property: for every \epsilon > 0 there exist measurable sets B and C such that :

    B \subset A \subset C and m(c \cap B^c) < \epsilon. Prove that A is measurable.

    thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Feb 2010
    Posts
    147
    Quote Originally Posted by davidmccormick View Post
    1. Suppose A is a bounded set and that m^{*}(<br />
A\cap I)\leq am^{*}(I) for every interval I and 0 < a < 1. Prove that m^{*}(A)=0. What if A is unbounded?

    2. A is a subset of \mathbb{R} with the following property: for every \epsilon > 0 there exist measurable sets B and C such that :

    B \subset A \subset C and m(c \cap B^c) < \epsilon. Prove that A is measurable.

    thanks.
    1.

    Let {  I_k } be a collection of open intervals such that  A \subset \bigcup I_k

    Now, note that  A = \bigcup_k (A \cap I_k) , so by subadditivity and the assumption we have:

     m^*(A) \leq \sum_k m^*((A \cap I_k) \leq a*\sum_k m^*(I_k) . I.e.

     \sum_k m^*(I_k) \geq \frac{m^*(A)}{a} for every collection of open intervals that cover A. Thus, we have that:

    inf{  \sum_k m^*(I_k) ;  A \subset \bigcup I_k }  \geq \frac{m^*(A)}{a} .

    But, the term on the left is m*(A)! Which gives us:

     m^*(A) \geq \frac{m^*(A)}{a} which is only possible in m*(A) = 0 (try it out)

    Notice that I never mentioned once that A is bounded

    EDIT: Monotonicity replaced with countable sub-additivity
    Last edited by southprkfan1; April 25th 2010 at 10:14 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Closure of a totaly bounded set is totally bounded
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: April 8th 2011, 07:42 PM
  2. Let f be a bounded measurable function on [a,b]...
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 6th 2010, 09:10 AM
  3. Lebesgue measurable function and Borel measurable function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 16th 2010, 02:55 AM
  4. Replies: 4
    Last Post: October 12th 2009, 08:43 PM
  5. Replies: 2
    Last Post: October 1st 2009, 07:07 PM

Search Tags


/mathhelpforum @mathhelpforum