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Thread: Bounded, measurable?

  1. #1
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    Bounded, measurable?

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

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

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

    thanks.
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  2. #2
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    Quote Originally Posted by davidmccormick View Post
    1. Suppose $\displaystyle A$ is a bounded set and that $\displaystyle m^{*}(
    A\cap I)\leq am^{*}(I) $ for every interval I and $\displaystyle 0 < a < 1$. Prove that $\displaystyle m^{*}(A)=0$. What if A is unbounded?

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

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

    thanks.
    1.

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

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

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

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

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

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

    $\displaystyle 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; Apr 25th 2010 at 10:14 AM.
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