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- Jun 22, 2012

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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

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 \(\displaystyle n = 1\) ... "

Can someone please demonstrate rigorously that 2.15 clearly implies 2.16 if \(\displaystyle n = 1\) ...

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

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

Hope that helps ...

Peter

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

Result 2.14 and its proof read as follows

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 \(\displaystyle n = 1\) ... "

Can someone please demonstrate rigorously that 2.15 clearly implies 2.16 if \(\displaystyle n = 1\) ...

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

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

Hope that helps ...

Peter

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