Given a Borel set B of finite Lebesgue measure show that for every ϵ &gt; 0, there exist: a compact set C 1 ⊆ B, a closed set C 2 ⊆ R ∖ B and a continuous function ϕ : R → [ 0 , 1 ] s.t.: χ C 1 ≤ ϕ ≤ χ R ∖ C 2 | | χ B − ϕ | | 1 &lt; ϵ"Soo I have only the vague intuition of using the monotone convergence theorem somehow - e.g. perhaps setting B ∩ [ n , n + 1 ] as sets on which corresponding simple functions f n (such as the indicator function) can be defined so that they are wedged between χ C 1 and χ R ∖ C 2 (thus satisfying the first condition) and so that taking the limit on their integral would satisfy the second condition.I am not sure how to go about accomplishing this, especially since the example I suggested for B ∩ [ n , n + 1 ] is not necessarily closed and therefore not necessarily compact. In addition, I am not sure how to define the required closed set C 2 .Any guidance on the steps of applying MCT (assuming I am at least right on this) to this problem would be much appreciated.Edit: Another thought I had was perhaps using somehow the regularity of the Lebesgue measure to define the compact and closed sets (?), i.e. there are a compact set K and an open set U such that: μ ( B ) ≤ μ ( K ) + ϵand μ ( B ) ≥ μ ( U ) − ϵMaybe these could be used to define the indicator function I mentioned above?

Question

Given a Borel set   B of finite Lebesgue measure show that for every   ϵ  &amp;gt;  0, there exist: a compact set       C    1    ⊆  B, a closed set       C    2    ⊆      R    ∖  B and a continuous function   ϕ  :      R    →  [  0  ,  1  ] s.t.:      χ                  C        1              ≤  ϕ  ≤      χ                  R            ∖              C        2                  |        |        χ    B    −  ϕ      |              |        1    &amp;lt;  ϵ&quot;Soo I have only the vague intuition of using the monotone convergence theorem somehow - e.g. perhaps setting   B  ∩  [  n  ,  n  +  1  ] as sets on which corresponding simple functions       f    n   (such as the indicator function) can be defined so that they are wedged between       χ                  C        1             and       χ                  R            ∖              C        2             (thus satisfying the first condition) and so that taking the limit on their integral would satisfy the second condition.I am not sure how to go about accomplishing this, especially since the example I suggested for   B  ∩  [  n  ,  n  +  1  ] is not necessarily closed and therefore not necessarily compact. In addition, I am not sure how to define the required closed set       C    2  .Any guidance on the steps of applying MCT (assuming I am at least right on this) to this problem would be much appreciated.Edit: Another thought I had was perhaps using somehow the regularity of the Lebesgue measure to define the compact and closed sets (?), i.e. there are a compact set   K and an open set   U such that:  μ  (  B  )  ≤  μ  (  K  )  +  ϵand  μ  (  B  )  ≥  μ  (  U  )  −  ϵMaybe these could be used to define the indicator function I mentioned above?

esperoanow · Accepted Answer

The problem is stated a bit weirdly. It is more commonly stated with   U  =      R    ∖      C    2  . To solve your problem, you can use regularity of Lebesgue measure to get   K and   U, and then use Urysohn&#039;s lemma to get   ϕ  ∈      C    c    (      R    ) (or even in       C    c          ∞        (      R    )) with   K  ≺  ϕ  ≺  U.

Given a Borel set B of finite Lebesgue measure show that for every ϵ &

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