Verifying pointwise convergence of indicator functions

  • #1
psie
282
33
TL;DR Summary
I'm stuck at something fairly basic I think. Let be an open ball of radius and center in . It is claimed that pointwise as and on , where is the sphere . I am stuck showing this.
I'm reading a proof of a lemma that where is Lebesgue measure, is jointly continuous in and ( stands for average). The claim that on is made in the proof. I think there are two cases to consider. Let .
  1. , i.e. . Is it then also true that for some sequences that converge to respectively, that for large enough ? Why? If yes, then for large enough too.
  2. Similarly, if , is it then true that ?
Also, why does on the sphere ?
 
Physics news on Phys.org
  • #2
My intuition says 1) yes, 2) possibly, and 3) because you cannot avoid the jump from to on the sphere as there is no neighborhood completely contained in either region. However, I have to consider the triangle inequalities in detail. I think the answer to 3) is the key to the first two questions.
 
  • Like
Likes psie
  • #3
fresh_42 said:
My intuition says 1) yes, 2) possibly, and 3) because you cannot avoid the jump from to on the sphere as there is no neighborhood completely contained in either region. However, I have to consider the triangle inequalities in detail. I think the answer to 3) is the key to the first two questions.
Ok. I agree, I don't see how to derive from the triangle inequality given and . It doesn't lead anywhere: Here for sufficiently large . But I don't see how else to show pointwise on .
 
  • #4
Actually, I worked out 1).

Let . For sufficiently large , and . So .
 
  • #5
psie said:
Ok. I agree, I don't see how to derive from the triangle inequality given and . It doesn't lead anywhere: Here for sufficiently large . But I don't see how else to show pointwise on .
We may assume w.l.o.g. that and We may also assume that where for some constant We then need the condition I struggle a bit to define appropriately such that and still holds, but that would be the idea.

The second is analogous with
 
  • Like
Likes psie

Similar threads

Replies
6
Views
1K
Replies
1
Views
177
Replies
3
Views
326
Replies
5
Views
2K
Replies
1
Views
292
Replies
2
Views
959
Replies
5
Views
839
Back
Top