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I'm having trouble starting two similar proofs:

Let $\epsilon > 0$. And let $E$ be a measurable set of finite measure.

Prove that there is an open set $U$ containing $E$ such that $m(U \setminus E) < \epsilon$.

Similarly, prove there is a compact set $K$ contained in $E$ such that $m(E \setminus K) < \epsilon$.

Any hints are much appreciated.

NOTE: $m$ is the Lebesgue outer measure. And $E \subseteq \mathbb{R}$ is measurable if $m(A) \geq m(A \cap E) + m(A \setminus E)$.

  • 2
    I presume $E$ is a subset of $\mathbb{R}$ or $\mathbb{R}^n$ and the measure is Lebesgue measure?2012-05-02
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    and that $m$ denotes Lebesgue measure?2012-05-02
  • 0
    The first assertive is kind of a definition the second one is the definition on the complementar of E.2012-05-02
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    And your definition of measurable set is?2012-05-02
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    Yes, $m$ is the Lebesgue measure. And $E \subseteq \mathbb{R}$ is measurable if $m(A) \geq m(A \cap E) + m(A \setminus E)$.2012-05-02
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    *Lebesgue _outer_ measure2012-05-02
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    @MikeC. Use the definition of outer measure and set $U$ = union of open intervals that cover $E$. Take complements to get the second part. I could say more if you want.2012-05-03
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    What is your definition of Lebesgue outer measure? Is it by using coverings of open intervals? something else?2012-06-08

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