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

[Math Amateur]

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:

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

 

[Jhoneine]

.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.

 

[soloenergy]

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.