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

abbracciopj 2022-06-30 Answered
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?
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Answers (2)

pressacvt
Answered 2022-07-01 Author has 19 answers
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.

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kokoszzm
Answered 2022-07-02 Author has 8 answers
Use your idea along with the property of "continuity from below" that all measures have. You can show that an event has measure one in this way.

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