Outer measure ... Axler, Result 2.14 ... Another Question ...

  • #1
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Summary:

I need further help in order to fully understand the proof that | [a, b] | = b - a ... ...
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...

I need further help with the proof of Result 2.14 ...

Result 2.14 and its proof read as follows


Axler - Result  2.14- outer measure of a closed interval .png





In the above proof by Axler we read the following:

" ... ... To get started with the induction, note that 2.15 clearly implies 2.16 if ##n = 1## ... "


Can someone please demonstrate rigorously that 2.15 clearly implies 2.16 if ##n = 1## ...

... in other words, demonstrate rigorously that ##[a, b] \subset I_1 \Longrightarrow l(I_1) \geq b - a## ...


My thoughts ... we should be able to use ##(a, b) \subset [a, b]## and the fact that if ##A \subset B## then ##\mid A \mid \leq \mid B \mid## ... ... but we may have to prove rigorously that ##\mid (a, b) \mid = b - a ## but how do we express this proof ...


Help will be much appreciated ... ...

Peter


=============================================================================================================

Readers of the above post may be assisted by access to Axler's definition of the length of an open interval and his definition of outer measure ... so I am providing access to the relevant text ... as follows:


Axler - Defn 2.1 & 2.2 .png






Hope that helps ...

Peter
 
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Answers and Replies

  • #2
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You start with ##[a,b]\subseteq I_1##. Write ##I_1=]c,d[##. Then ##l(I_1)= d-c\geq b-a##. Not sure if that solves your problem?
 
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  • #3
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You start with ##[a,b]\subseteq I_1##. Write ##I_1=]c,d[##. Then ##l(I_1)= d-c\geq b-a##. Not sure if that solves your problem?

I believe that does solve the problem ...

Just a supplementary question ... Axler only defines the length of an open interval ... he has now proven that ##\mid [a, b] \mid = b - a## ...

... however, nothing has been said about the length of ## [a, b] ## ... do we know what ##l( [a, b] )## is ... ?

How would we show rigorously that ##l( [a, b] ) = b - a ...?

I note in passing that other books approach this issue by defining ##l( [a, b] ) = l( (a, b) ) = b - a## ...



... and another supplementary question ...



How would we show rigorously that ## \mid (a, b) \mid = b - a## ....

I know it seems intuitively obvious but how would you express a convincing and rigorous proof of the above result ...


Peter
 
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  • #4
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I believe that does solve the problem ...

Just a supplementary question ... Axler only defines the length of an open interval ... he has now proven that ##\mid [a, b] \mid = b - a## ...

... however, nothing has been said about the length of ## [a, b] ## ... do we know what ##l( [a, b] )## is ... ?

How would we show rigorously that ##l( [a, b] ) = b - a ...?

I note in passing that other books approach this issue by defining ##l( [a, b] ) = l( (a, b) ) = b - a## ...



... and another supplementary question ...



How would we show rigorously that ## \mid (a, b) \mid = b - a## ....

I know it seems intuitively obvious but how would you express a convincing and rigorous proof of the above result ...


Peter
Axler defined the length ##l## only for open intervals, so it does not make sense to ask what ##l([a,b])## is without giving an appropriate definition.

You proved that ##|[a,b]| = b-a##. Similarly, you can prove that ##|(a,b)| = b-a##. So when you will continue reading, you will note that the function ##A \mapsto |A|## restricted to "good" subsets has properties you want a length to have. So actually what you are doing is constructing a measure on some collection of subsets of the reals such that it extends the length of the open interval ##(a,b)## in an intuitive and good way. Hope this helps.
 
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  • #5
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Axler defined the length ##l## only for open intervals, so it does not make sense to ask what ##l([a,b])## is without giving an appropriate definition.

You proved that ##|[a,b]| = b-a##. Similarly, you can prove that ##|(a,b)| = b-a##. So when you will continue reading, you will note that the function ##A \mapsto |A|## restricted to "good" subsets has properties you want a length to have. So actually what you are doing is constructing a measure on some collection of subsets of the reals such that it extends the length of the open interval ##(a,b)## in an intuitive and good way. Hope this helps.
Thanks ... yes definitely helps a lot ...

Still reflecting on what you have written...

Thanks again...

Peter
 

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