I am a graduate student of Mathematics.In the book Measure,Integration and Real Analysis by Sheldon Axler there is a question that asks the reader to show that there exists no measure space ( X , S , μ ) such that { μ ( E ) : E ∈ S } = [ 0 , 1 ).I am not sure how to do it.I was thinking of taking a nested sequence { E n } of sets in S such that μ ( E n ) = 1 − 1 / n but I don't think that will work.Can someone give me any clue?

Question

I am a graduate student of Mathematics.In the book Measure,Integration and Real Analysis by Sheldon Axler there is a question that asks the reader to show that there exists no measure space   (  X  ,      S    ,  μ  ) such that   {  μ  (  E  )  :  E  ∈      S    }  =  [  0  ,  1  ).I am not sure how to do it.I was thinking of taking a nested sequence   {      E    n    } of sets in       S   such that   μ  (      E    n    )  =  1  −  1      /    n but I don&#039;t think that will work.Can someone give me any clue?

pressacvt · Accepted Answer

If   (  X  ,      S    ,  μ  ) denotes a measure space then       S   denotes a   σ-algebra on   X so that by definition   X  ∈      S   and consequently:  μ  (  X  )  ∈  {  μ  (  E  )  ∣  E  ∈      S    }This with   μ  (  E  )  ≤  μ  (  X  ) for every   E  ∈      S  .So the set   {  μ  (  E  )  ∣  E  ∈      S    } has a largest element in   μ  (  X  ).However evidently the set   [  0  ,  1  ) does not have a largest element and we conclude that the two sets do not coincide.

kokoszzm · Accepted Answer

Use your idea along with the property of &quot;continuity from below&quot; that all measures have. You can show that an event has measure one in this way.

I am a graduate student of Mathematics.In the book Measure,Integration and Real Analysis by Sheldon

Answered question

Answer & Explanation

New Questions in Pre-Algebra