Lebesgue Outer Measure .... Carothers, Proposition 16.1 ....

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In summary, N. L. Carothers' book: "Real Analysis" discusses the Lebesgue measure. He suggests that the $J_k$ be expanded and the $I_n$ be shrunk, and that this alteration of the intervals preserves the inclusion $\bigcup_{n=1}^{N}I_{n}'\subset \bigcup_{k=1}^{\infty}J_{k}$. However, this alteration also preserves the inequality of the summations.
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I am reading N. L. Carothers' book: "Real Analysis". ... ...

I am focused on Chapter 16: Lebesgue Measure ... ...

I need help with an aspect of the proof of Proposition 16.1 ...

Proposition 16.1 and its proof read as follows:
Carothers - Proposition 16.1 ... .png

In the above text from Carothers we read the following:

" ... ... But now, by expanding each $J_k$ slightly and shrinking each $I_n$ slightly, we may suppose that the $J_k$ are open and the $I_n$ are closed. ... "Can someone please explain how Carothers is expecting the $J_k$ to be expanded and the $I_n$ to be shrunk ... and further, why the proof is still valid after the $J_k$ and $I_n$ have been altered in this way ... ...
EDIT: My thoughts ...

We could expand each $J_k$ by altering or replacing intervals of the form $(a, b), [a, b)$ and $(a, b]$ by $[a, b]$ ...

This would expand the $J_k$ by one or two points only leaving the length of the intervals unchanged ...

BUT ... we cannot (as Carothers wishes) then suppose the $J_k$ are open ... indeed they would all be closed ... so we have to find another to expand the $J_k$ slightly

A similar problem arises if we shrink the I_n by replacing intervals of the form $[a, b], [a, b)$ and $(a, b]$ by $(a, b)$ ...

Help will be appreciated ...

Peter
 
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Hi Peter,

This is a sneaky argument made by the author. Let $D = \sum_{n=1}^{N} l(I_{n}) - \sum_{k=1}^{\infty}l(J_{k})>0$. Let $\delta = D/8$. Expand each $J_{k}$ to an open interval, say $J_{k}'$, of length $l(J_{k}) + \delta/2^{k}$ and shrink each $I_{n}$ to a closed interval, say $I_{n}'$, of length $l(I_{n}) - \delta/2^{n}$. Note that this alteration of the intervals preserves the inclusion $\bigcup_{n=1}^{N}I_{n}'\subset \bigcup_{k=1}^{\infty}J_{k}'.$ It also preserves the inequality of the summations. Indeed, by the way $\delta$ was chosen, $$\sum_{k=1}^{\infty}l(J_{k}') = \sum_{k=1}^{\infty}l(J_{k}) + \delta < \sum_{n=1}^{N}l(I_{n}) - \delta =\sum_{n=1}^{N}l(I_{n}) - \sum_{n=1}^{\infty}\delta/2^{n} < \sum_{n=1}^{N}\left[l(I_{n})-\delta/2^{n}\right] = \sum_{n=1}^{N}l(I_{n}').$$ Hence, both the set inclusion and the inequality in the summation remain the same, even when the specified alterations are made to $I_{n}$ and $J_{k}$. This is a tricky one, so certainly feel free to let me know if anything is still unclear.

Edit: The length of $I_{n}'$ should be $l(I_{n}) - \delta/2^{n}$. There was a typo in my original post when I wrote $l(I_{n})+\delta/2^{n}$.
 
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  • #3
Thanks for a most helpful reply GJA ...

I would never have seen that without your help!

Indeed, I am still studying and reflecting on your post ...

Peter
 

Related to Lebesgue Outer Measure .... Carothers, Proposition 16.1 ....

1. What is Lebesgue Outer Measure?

Lebesgue Outer Measure is a mathematical concept used in measure theory to assign a numerical value to sets in a given space. It is defined as the infimum of the sum of the lengths of intervals that cover a given set.

2. How is Lebesgue Outer Measure calculated?

The Lebesgue Outer Measure of a set is calculated by taking the infimum of the sum of the lengths of intervals that cover the set. This means finding the smallest possible value that can cover the set.

3. What is Proposition 16.1 in Carothers' book?

Proposition 16.1 in Carothers' book is a mathematical theorem that states that the Lebesgue Outer Measure of a set is equal to the Lebesgue Measure of the set, if the set is measurable. In other words, if a set can be accurately measured, its Lebesgue Outer Measure is equal to its Lebesgue Measure.

4. What is the significance of Proposition 16.1 in Carothers' book?

Proposition 16.1 is significant because it provides a way to calculate the Lebesgue Outer Measure of a set by using its Lebesgue Measure. This allows for a more precise and accurate calculation of the measure of a set, which is important in various areas of mathematics such as analysis and probability theory.

5. How is Lebesgue Outer Measure used in mathematics?

Lebesgue Outer Measure is used in mathematics to measure the size or extent of sets in a given space. It is an important concept in measure theory and is used in various areas of mathematics such as analysis, probability theory, and geometry. It allows for a more precise and accurate measurement of sets, which is essential in many mathematical applications.

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