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

  • #1
Math Amateur
Gold Member
1,067
47

Summary:

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



Help will be appreciated ...

Peter
 

Answers and Replies

  • #2
Math_QED
Science Advisor
Homework Helper
2019 Award
1,693
719
Replace ##J_k## by its closure and ##I_n## by its interior. These operations add/remove only begin or end points of the intervals so the lengths of the intervals are unaffected and the inequality with the two series keep valid.
 
  • Like
Likes Math Amateur
  • #3
Math Amateur
Gold Member
1,067
47
Replace ##J_k## by its closure and ##I_n## by its interior. These operations add/remove only begin or end points of the intervals so the lengths of the intervals are unaffected and the inequality with the two series keep valid.
Thanks for the help ...

I get the idea but shouldn't we replace ##J_k## by its interior and ##I_n## by its closure ...

Peter
 
  • #4
Math Amateur
Gold Member
1,067
47
You write:

" ... Interior makes the set smaller, the closure of the set makes the set larger. ... "

Yes ... but the problem I have is that we are told we may suppose that the ##J_k## are open and the ##I_n## are closed!

If we replace ##J_k## by its closure the ##J_k## will be closed not open ...

Peter
 
  • #5
13,457
10,517
You write:

" ... Interior makes the set smaller, the closure of the set makes the set larger. ... "

Yes ... but the problem I have is that we are told we may suppose that the ##J_k## are open and the ##I_n## are closed!

If we replace ##J_k## by its closure the ##J_k## will be closed not open ...

Peter
You can quantify it if you like. As mentioned by @Math_QED we need only a point or two from not closed to closed, which doesn't change the length. And we need an arbitrary small, but positive distance to change from closed to open.

If there is a strict inequality, then it has a positive distance ##d>0## between the two. So the inequality still holds if we e.g. add ##d/2## to the smaller sum. Now we can write ##d/2=\sum_{n=1}^\infty d\left(\dfrac{1}{3}\right)^n##. This means we have ##(1/2)d/3^k## available to add on each side of every interval ##J_k## to make it open and still have a total length strictly smaller than ##d##.
 
Last edited:
  • Like
Likes Math Amateur and Math_QED
  • #6
Math Amateur
Gold Member
1,067
47
Thanks for your assistance fresh_42 ...

Your post was extremely helpful...

Peter
 

Related Threads on Lebesgue Outer Measure ... Carothers, Proposition 16.1 ...

Replies
6
Views
235
  • Last Post
Replies
2
Views
1K
Replies
6
Views
826
Replies
14
Views
334
Replies
8
Views
2K
Replies
12
Views
3K
Replies
4
Views
970
Replies
16
Views
2K
Replies
2
Views
1K
Top