Translation Invariance of Outer Measure ... Axler, Result 2.7 ...

  • #1
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Summary:

I need help in order to fully understand Axler's proof of the translation invariance of outer measure ...
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...

I need help with the proof of Result 2.7 ...

Result 2.7 and its proof read as follows:


Axler - Result  2.7 - outer measure is translation invariant .png





In the above proof by Axler we read the following:

" ... ... Thus

... ##\mid t + A \mid \leq \sum_{ k = 1 }^{ \infty } l ( t + I_k ) = \sum_{ k = 1 }^{ \infty } l ( I_k )##

Taking the infimum of the last term over all sequences ##I_1, I_2, ... ## of open intervals whose union contains ##A##, we have ##\mid t + A \mid \leq \mid A \mid##. ... ..."


Can someone please explain exactly how/why taking the infimum of the last term over all sequences ##I_1, I_2, ... ## of open intervals whose union contains ##A##, we have ##\mid t + A \mid \leq \mid A \mid## ... ?...


Peter
 

Answers and Replies

  • #2
Math_QED
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Use a similar lemma as in your previous questions:

If ##A## is a non-emptyset of ##\Bbb{R}## and ##b## is a real number with ##b\leq a## for all ##a\in A##, then ##b\leq \inf(A)##.
 
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  • #3
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Thanks Math_QED ...
 
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