Outer measure of a closed interval ... Axler, Result 2.14 ...

  • #1
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Summary:

I need 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 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:

" ... ... We will now prove by induction on n that the inclusion above implies that


## \sum_{ k = 1 }^n l(I_k) \ \geq b - a##


This will then imply that ##\sum_{ k = 1 }^{ \infty } l(I_k) \geq \sum_{ k = 1 }^n l(I_k) \ \geq b - a##, completing the proof that ##\mid [a, b] \mid \ \geq b - a##. ... ... "


Can someone please explain exactly why ##\sum_{ k = 1 }^{ \infty } l(I_k) \ \geq \sum_{ k = 1 }^n l(I_k) \ \geq b - a## completes the proof that ##\mid [a, b] \mid \ \geq b - a##. ... ...

Indeed ... can someone please show, formally and rigorously, that ##\sum_{ k = 1 }^{ \infty } l(I_k) \ \geq \sum_{ k = 1 }^n l(I_k) \ \geq b - a## implies that ##\mid [a, b] \mid \geq b - a##. ... ...



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
 

Answers and Replies

  • #2
Math_QED
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The author uses the following fact: Let ##A## be a non-empty set and ##b## be a fixed number. If ##a \geq b## for all ##a \in A##, then ##\inf(A) \geq b## (immediate from the definition of infinum: ##b## is a lower bound of ##A##).

In your case, the author shows ##b-a \leq |[a,b]|## by showing that ##b-a \leq \sum_k l(I_k)## for all choices of ##I_k## as in the definition of outer measure.
 
  • #3
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The author uses the following fact: Let ##A## be a non-empty set and ##b## be a fixed number. If ##a \geq b## for all ##a \in A##, then ##\inf(A) \geq b## (immediate from the definition of infinum: ##b## is a lower bound of ##A##).

In your case, the author shows ##b-a \leq |[a,b]|## by showing that ##b-a \leq \sum_k l(I_k)## for all choices of ##I_k## as in the definition of outer measure.

Thanks Math_QED ... appreciate your help ...

Still reflecting on what you have written ...

Peter
 

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