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

In summary, Peter is trying to figure out the proof for Result 2.14 in Sheldon Axler's book: Measure, Integration & Real Analysis. He needs help understanding why the infimum of $\left\{\sum_{k=1}^{\infty}l(I_{k})\,:\, I_{1}, I_{2},\ldots \text{ are open intervals such that } A\subset\bigcup_{k=1}^{\infty}I_{k} \right\}$ is equal to b-a.
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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 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 \(\displaystyle \sum_{ k = 1 }^n l(I_k) \ \geq b - a\)This will then imply that \(\displaystyle \sum_{ k = 1 }^{ \infty } l(I_k) \geq \sum_{ k = 1 }^n l(I_k) \ \geq b - a\), completing the proof that \(\displaystyle \mid [a, b] \mid \ \geq b - a\). ... ... "Can someone please explain exactly why \(\displaystyle \sum_{ k = 1 }^{ \infty } l(I_k) \ \geq \sum_{ k = 1 }^n l(I_k) \ \geq b - a\) completes the proof that \(\displaystyle \mid [a, b] \mid \ \geq b - a\). ... ...

Indeed ... can someone please show, formally and rigorously, that \(\displaystyle \sum_{ k = 1 }^{ \infty } l(I_k) \ \geq \sum_{ k = 1 }^n l(I_k) \ \geq b - a\) implies that \(\displaystyle \mid [a, b] \mid \geq b - a\). ... ...
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
 
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  • #2
Hi Peter,

According to the author, it has been established that $\vert [a,b]\vert \leq b-a.$ Hence, it only remains to show that $\vert [a,b]\vert \geq b-a.$ Since $\{I_{k}\}_{k=1}^{\infty}$ is an arbitrary collection of open intervals that covers $[a,b]$, this proof establishes the fact that $b-a$ is a lower bound for the set $\left\{\sum_{k=1}^{\infty}l(I_{k})\,:\, I_{1}, I_{2},\ldots \text{ are open intervals such that } A\subset\bigcup_{k=1}^{\infty}I_{k} \right\}.$ Since the infimum is the greatest lower bound, it follows that $$b-a\leq\inf{\left\{\sum_{k=1}^{\infty}l(I_{k})\,:\, I_{1}, I_{2},\ldots \text{ are open intervals such that } A\subset\bigcup_{k=1}^{\infty}I_{k} \right\}} =\vert[a,b]\vert\leq b-a.$$
 
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  • #3
Thanks GJA ...

Really appreciate your help ...

Peter
 

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

1. What is the outer measure of a closed interval?

The outer measure of a closed interval is the length of the interval. It is defined as the smallest number that is greater than or equal to the sum of the lengths of any countable collection of open intervals that cover the closed interval.

2. How is the outer measure of a closed interval calculated?

The outer measure of a closed interval can be calculated by taking the length of the interval. For example, if the closed interval is [a,b], the outer measure would be b-a.

3. What is Result 2.14 in relation to the outer measure of a closed interval?

Result 2.14 is a theorem in mathematics that states that the outer measure of a closed interval is equal to its length. This result is important because it allows us to calculate the outer measure of a closed interval using its length.

4. Why is the outer measure of a closed interval important?

The outer measure of a closed interval is important because it helps us understand the concept of measure in mathematics. It allows us to calculate the size or length of a closed interval, which is useful in many mathematical applications.

5. How does the outer measure of a closed interval relate to the concept of measure in mathematics?

The outer measure of a closed interval is a specific type of measure in mathematics. It is a way of assigning a number to a set that represents its size or length. The outer measure is a fundamental concept in measure theory, which is a branch of mathematics that deals with the study of measures.

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