a saw some proofs of Lebesgue regularity and open set G (for E ∈ L ) such that λ ( G ∖ E ) &lt; ε, but how to proof this one: Namely suppose E ∈ L is such that for ε &gt; 0 there exist a closed set F ⊂ E such that λ ( E ∖ F ) &lt; ε?

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a saw some proofs of Lebesgue regularity and open set G (for E  ∈      L  ) such that   λ  (  G  ∖  E  )  &amp;lt;  ε, but how to proof this one: Namely suppose   E  ∈      L   is such that for   ε  &amp;gt;  0 there exist a closed set   F  ⊂  E such that   λ  (  E  ∖  F  )  &amp;lt;  ε?

Jayvion Tyler · Accepted Answer

E  ∈      L      ⟹        E    c    ∈      L  Now, given any   ϵ  &amp;gt;  0 ,   ∃  G open set such that       E    c    ⊂  G and   λ  (  G  ∖      E    c    )  &amp;lt;  ϵAs,       E    c    ⊂  G    ⟹        G    c    ⊂  EThen,       G    c    =  F is a closed set and  λ  (  G  ∖      E    c    )  =  λ  (  E  ∖  F  )  &amp;lt;  ϵ

a saw some proofs of Lebesgue regularity and open set G (for E ∈ <mrow class=

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