I'm working through some literature from Geometric Measure Theory for an assignment paper and have got stuck at a step in the proof of the Federer-Volpert theorem. The reasoning is as follows: according to the BV coarea formula, the level sets { u &gt; t } have finite perimeter for λ-a.e. t ∈ R . From this, the author concludes that we can find a dense subset D of { t ∈ R | { u &gt; t } has finite perimeter }. I'm guessing this must follow from some well-known fact in measure theory, but I can't put my finger on it. Any help would be greatly appreciated.

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I&#039;m working through some literature from Geometric Measure Theory for an assignment paper and have got stuck at a step in the proof of the Federer-Volpert theorem. The reasoning is as follows: according to the BV coarea formula, the level sets   {  u  &amp;gt;  t  } have finite perimeter for   λ-a.e.   t  ∈      R  . From this, the author concludes that we can find a dense subset   D of   {  t  ∈      R       |     {  u  &amp;gt;  t  }     has finite perimeter  }. I&#039;m guessing this must follow from some well-known fact in measure theory, but I can&#039;t put my finger on it. Any help would be greatly appreciated.

trantegisis · Accepted Answer

If   P  (  t  ) is any property that holds for almost every   t  ∈      R  , then   {  t  ∈      R    :  P  (  t  )  } is dense in       R  .Proof. By assumption,   λ  (  {  t  ∈      R    :  not     P  (  t  )  }  )  =  0, and hence   {  t  ∈      R    :  not     P  (  t  )  } contains no nontrivial interval   (  a  ,  b  ). Therefore,   {  t  ∈      R    :  P  (  t  )  } intersects every nontrivial interval   (  a  ,  b  ), so it is dense.

I'm working through some literature from Geometric Measure Theory for an assignment paper and have g

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