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

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In summary: Your Name]In summary, Peter.Carothers is suggesting to expand the J_k and shrink the I_n in order to satisfy the necessary conditions for the proof of Proposition 16.1 in his book "Real Analysis". This is done by adding a small portion of the surrounding area to the J_k and removing a small portion of the area contained in I_n. The slight alterations do not affect the validity of the proof, as they do not significantly change the properties of the intervals.
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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 \(\displaystyle J_k\) slightly and shrinking each \(\displaystyle I_n\) slightly, we may suppose that the \(\displaystyle J_k\) are open and the \(\displaystyle I_n\) are closed. ... "Can someone please explain how Carothers is expecting the \(\displaystyle J_k\) to be expanded and the \(\displaystyle I_n\) to be shrunk ... and further, why the proof is still valid after the \(\displaystyle J_k\) and \(\displaystyle I_n\) have been altered in this way ... ...
Help will be appreciated ...

Peter
 
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.Carothers is suggesting that you expand each J_k by adding a small portion of the surrounding area to it, and shrink each I_n by removing a small portion of the area it contains. This ensures that the J_k are open sets (i.e. they contain all their boundary points) and the I_n are closed sets (i.e. they contain none of their boundary points). This does not affect the validity of the proof, because the expansion and shrinking of the J_k and I_n do not affect the measure of the set A, which is what Proposition 16.1 is concerned with.
 
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Hello Peter,

In order to understand how Carothers is suggesting to expand the J_k and shrink the I_n, let's first look at the context of Proposition 16.1. The proposition states that for a finite or countable collection of intervals I_n, the Lebesgue measure of their union is equal to the sum of their individual measures. In the proof, Carothers uses the fact that any open set can be written as a countable union of closed intervals. This is where the J_k and I_n come into play.

Now, in order for the J_k to be open and the I_n to be closed, they need to satisfy certain conditions. For example, the J_k may need to be expanded slightly in order to be open, and the I_n may need to be shrunk slightly in order to be closed. By doing so, Carothers is making sure that the J_k and I_n satisfy the necessary conditions for the proof to hold.

To understand why the proof is still valid after this alteration, we need to keep in mind that the expansion and shrinking are done only slightly, and they do not change the properties of the intervals significantly. Therefore, the proof still holds because the J_k and I_n are still essentially the same intervals, just with slight alterations to satisfy the necessary conditions.

I hope this helps clarify the concept for you. Let me know if you have any further questions.
 

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

1. What is Lebesgue Outer Measure?

Lebesgue Outer Measure is a measure of the size or length of a set in a given space, defined by French mathematician Henri Lebesgue in the early 20th century. It is a generalization of the concept of length or volume to more complex sets, and is used in measure theory to study the properties of sets and functions.

2. How is Lebesgue Outer Measure calculated?

Lebesgue Outer Measure is calculated using the concept of outer measure, which is defined as the infimum of the sum of the lengths of intervals that cover a given set. In simpler terms, it is the smallest possible length that can cover a set. This calculation method allows for the measure to be extended to more complex sets, such as fractals or irregular shapes.

3. What is the significance of Proposition 16.1 in Carothers' work on Lebesgue Outer Measure?

Proposition 16.1 in Carothers' work states that the outer measure of a set can be approximated by the sum of the outer measures of a sequence of simpler sets. This is important because it allows for the calculation of outer measure for more complex sets, and is a key step in the development of the Lebesgue measure theory.

4. How is Lebesgue Outer Measure different from other measures?

Lebesgue Outer Measure differs from other measures, such as the Riemann integral, in that it can be defined for more complex sets and is not limited to just intervals. It also has the important property of countable subadditivity, which means that the measure of the union of countably many sets is equal to the sum of their individual measures.

5. What are the practical applications of Lebesgue Outer Measure?

Lebesgue Outer Measure has many practical applications, particularly in the fields of probability and statistics. It is used to define the Lebesgue integral, which is a more general and powerful way to calculate integrals compared to the Riemann integral. It is also used in the study of fractals and other complex shapes, and has applications in physics, economics, and other fields.

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