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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 $$n = 1$$ ... "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:
Hope that helps ...
Peter
I need further help with the proof of Result 2.14 ...
Result 2.14 and its proof read as follows
" ... ... 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:
Hope that helps ...
Peter
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