# Outer Measure .... Axler, Result 2.5 ....

• MHB
• Math Amateur
In summary, the proof of Result 2.5 and its proof reads as follows:Now \mid A \mid \leq \sum_{ k = 1 }^{ \infty } l(I_k) follows from Axler's definition of outer measure ( is that correct?) ... see definition below ...Then essentially we have to prove that \mid A \mid \leq \text{ inf } ( \sum_{ k = 1 }^{ \infty } l(I_k) \text{ where } B \subset \cup_{ k = 1 }^{ \infty } I_k ) But how do we rigorously prove this ...Can someone
Math Amateur
Gold Member
MHB
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 1: Measures ...

I need help with the proof of Result 2.5 ...

Result 2.5 and its proof read as follows:

Now $$\displaystyle \mid A \mid \leq \sum_{ k = 1 }^{ \infty } l(I_k)$$ follows from Axler's definition of outer measure ( is that correct?) ... see definition below ...

Then essentially we have to prove that $$\displaystyle \mid A \mid \leq \sum_{ k = 1 }^{ \infty } l(I_k) \Longrightarrow \mid A \mid \leq \text{ inf } ( \sum_{ k = 1 }^{ \infty } l(I_k) \text{ where } B \subset \cup_{ k = 1 }^{ \infty } I_k )$$ ...

But how do we rigorously prove this ...

Can someone please demonstrate a formal and rigorous proof that:

$$\displaystyle \mid A \mid \leq \sum_{ k = 1 }^{ \infty } l(I_k) \Longrightarrow \mid A \mid \leq \text{ inf } ( \sum_{ k = 1 }^{ \infty } l(I_k) \text{ where } B \subset \cup_{ k = 1 }^{ \infty } I_k )$$ ...Help will be much appreciated ...===============================================My thoughts ...

Perhaps we can assume that $$\displaystyle \mid A \mid \ \gt \text{ inf } ( \sum_{ k = 1 }^{ \infty } l(I_k) \text{ where } B \subset \cup_{ k = 1 }^{ \infty } I_k )$$ ... and obtain a contradiction ...

We have that $$\displaystyle \mid A \mid \ \gt \text{ inf } ( \sum_{ k = 1 }^{ \infty } l(I_k) \text{ where } B \subset \cup_{ k = 1 }^{ \infty } I_k ) \Longrightarrow \ \exists \ \sum_{ k = 1 }^{ \infty } l(I_k)$$ such that $$\displaystyle \mid A \mid \ \gt \sum_{ k = 1 }^{ \infty } l(I_k)$$ where $$\displaystyle \sum_{ k = 1 }^{ \infty } I_k$$ covers $$\displaystyle B$$ ... Is this a contradiction ...? why exactly? How would you explain the contradiction clearly and rigorously ...

Hope that someone can help ...

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

Last edited:
Hi Peter,

You're almost there. Note that the collection of intervals is also a cover of $A$. Can you arrive at your contradiction using this fact?

GJA said:
Hi Peter,

You're almost there. Note that the collection of intervals is also a cover of $A$. Can you arrive at your contradiction using this fact?
Thanks for the help GJA ...

I think I can proceed now ... as follows ...

Assume that $$\displaystyle \mid A \mid \ \gt \text{ inf } ( \sum_{ k = 1 }^{ \infty } l(I_k) \text{ where } B \subset \cup_{ k = 1 }^{ \infty } I_k )$$

... then $$\displaystyle \ \exists \$$ a particular covering $$\displaystyle I_1, I_2, I_3,$$ ... of $$\displaystyle B$$ such that $$\displaystyle \mid A \mid \ \gt \sum_{ k = 1 }^{ \infty } l(I_k)$$ where $$\displaystyle B \subset \cup_{ k = 1 }^{ \infty } I_k$$

But this particular covering also covers $$\displaystyle A$$ since $$\displaystyle A \subset B$$ ...

... so that $$\displaystyle \mid A \mid \ \leq \sum_{ k = 1 }^{ \infty } l(I_k)$$ ... since $$\displaystyle \mid A \mid$$ is a lower bound on $$\displaystyle \sum_{ k = 1 }^{ \infty } l(I_k)$$ for all coverings $$\displaystyle I_1, I_2, I_3,$$ ... of $$\displaystyle A$$

... BUT ... $$\displaystyle \mid A \mid \ \leq \sum_{ k = 1 }^{ \infty } l(I_k)$$ is a contradiction of our assumption ...
Is the above correct?

Can someone please critique the above argument ...

Help will be appreciated ...

Peter

Looks great, Peter. Nicely done!

Thanks GJA ... appreciate your help as usual...

Peter

## 1. What is Outer Measure in mathematics?

Outer Measure is a concept in mathematics that measures the size or extent of a set. It is a generalization of the notion of length, area, or volume to sets that may not have a well-defined geometric shape.

## 2. How is Outer Measure calculated?

The Outer Measure of a set is calculated by taking the infimum (greatest lower bound) of the sum of the lengths of a countable collection of intervals that cover the set. In simpler terms, it is the smallest possible sum of the lengths of intervals that cover the set.

## 3. What is the significance of Outer Measure in measure theory?

Outer Measure is an important concept in measure theory as it helps to define the notion of measurable sets. A set is considered measurable if its Outer Measure equals its Inner Measure, which is defined as the supremum (least upper bound) of the sum of the lengths of a countable collection of intervals contained within the set.

## 4. What is the difference between Outer Measure and Lebesgue Measure?

Outer Measure is a more general concept than Lebesgue Measure. While Outer Measure can be defined for any set, Lebesgue Measure is only defined for measurable sets. Additionally, Outer Measure is a pre-measure, meaning it does not satisfy all the properties of a measure, whereas Lebesgue Measure is a complete measure.

## 5. How is Outer Measure used in real-world applications?

Outer Measure has many practical applications, particularly in the fields of physics, engineering, and economics. It is used to define the concept of length, area, and volume for sets that do not have a well-defined geometric shape, such as fractals. It is also used in the study of probability and statistics to define the probability of events that cannot be measured precisely.

Replies
2
Views
2K
Replies
2
Views
942
Replies
2
Views
969
Replies
2
Views
2K
Replies
2
Views
2K
Replies
8
Views
2K
Replies
3
Views
1K
Replies
7
Views
2K
Replies
2
Views
2K
Replies
2
Views
628