Measure of borel set minus open <e

In summary, the student is trying to figure out how to chop up an open set so that its measure is less than \epsilon . He is using countable union, intersection, and complement of open sets to try and find a solution. It turns out that if the open set is generated by some of the Xk 's, then the measure of U is also generated by those same Xk 's.
  • #1
ArcanaNoir
779
4

Homework Statement


We have a metric space [tex] X=\cup X_k [/tex] where [itex] X_k\subset X_{k+1}[/itex] and each [itex] X_k[/itex] is open. Show that for any Borel set E, there is an open set U such that [itex] \mu (U-E)<\epsilon [/itex]. (Its supposed to be "U \ E".)

Homework Equations



[itex] \mu [/itex] is a measure, so probably the important thing is countable subadditivity.
A borel set is a set generated by countable union, countable intersection, and relative complement of open sets.

The Attempt at a Solution



I know that if I have an open set I can intersect it with an [itex] X_k[/itex] and still have an open set... In this way I believe I can chop up any open set to countable pieces. But how can I get the difference in measure to be less than [itex] \epsilon [/itex] ?
The only solutions to a problem like this that I have seen are in the context of Lebesgue measure and [itex]\mathbb{R}^n [/itex], but I cannot use this context. I must prove it in a general metric space with a general measure. Also, if anyone can recommend a book that has a good treatment of general measures instead of focusing on Lebesgue, I would appreciate the suggestion. So far I have Royden and Folland.
 
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  • #2
Going back to set notation A/B= A[itex]\cap[/itex] B* (with * denoting the compliment).

This should make it easier to use like like countable subadditivity and finite intersections for measure.
 
  • #3
Okay, good point, I like that. But... I still don't know how to get an open set (or countably many open sets) tightly wrapped around E.
 
  • #4
ArcanaNoir said:
Okay, good point, I like that. But... I still don't know how to get an open set (or countably many open sets) tightly wrapped around E.

You're working in a metric space so you should consider looking at a collection (or more than one collection) of open balls.
 
  • #5
I'm just guessing because I haven't actually thought this problem out in detail but...

If we let E be generated by some of the Xk 's. I would think for a general borel set it doesn't have to be the X's but I'm not sue if the logic would flow nicely otherwise.


Then E= [itex]\cup[/itex]some k's Xk.
So E*= ([itex]\cup[/itex]k Xk)* = [itex]\cap[/itex] Xk*

and U/E is simply U[itex]\cap[/itex]([itex]\cap[/itex]k Xk) so this leaves you with a countable number of intersections.

And then you can try and work it from here. I think you may have to assume at some point that X has finite measure; not sure though.
 
  • #6
Jufro said:
If we let E be generated by some of the Xk 's. I would think for a general borel set it doesn't have to be the X's but I'm not sue if the logic would flow nicely otherwise.Then E= [itex]\cup[/itex]some k's Xk.
So E*= ([itex]\cup[/itex]k Xk)* = [itex]\cap[/itex] Xk*

and U/E is simply U[itex]\cap[/itex]([itex]\cap[/itex]k Xk) so this leaves you with a countable number of intersections.

But what if E is not a union of Xk's? What if E is only parts of the Xks?

And then you can try and work it from here. I think you may have to assume at some point that X has finite measure; not sure though.
Right, I forgot to mention [itex] \mu (X_k)<\infty [/itex]
 
  • #7
pasmith said:
You're working in a metric space so you should consider looking at a collection (or more than one collection) of open balls.

Yeah, but how would I get only countably many open balls?
 
  • #8
Yeah, generating from the Xk 's is kind of nice but isn't really true.

But that is ok because if your generating sets are Ji then each Ji [itex]\subset[/itex] [itex]\cup[/itex] for some k's Xk.

And then you could use that the fact that if A[itex]\subset[/itex] B then μ(A)≤ μ(B).

I think that should be enough to get the ball rolling.

The goal, in my mind, is to find what the measure of set U has to be and so that you can try and construct it.

Also, a trivial answer is U = ∅. Cheap, but finding a solution sort of proves that a solution has to exist.
 
Last edited:
  • #9
Holy crap I solved it.
 

1. What is a Borel Set?

A Borel set is a set of real numbers that can be constructed using a countable number of open or closed intervals. It is named after French mathematician Emile Borel who introduced the concept in the early 20th century.

2. What is the Measure of a Borel Set?

The measure of a Borel set is the length, area, or volume of the set. It represents the size or extent of the set in mathematical terms.

3. What is an Open Set?

An open set is a set that does not contain its boundary points. In other words, every point in the set has a small neighborhood around it that is also contained in the set. Examples of open sets include intervals, circles, and spheres.

4. What is the Difference Between a Borel Set and an Open Set?

The main difference between a Borel set and an open set is that a Borel set can be constructed using a countable number of open or closed intervals, while an open set is defined by not containing its boundary points. All Borel sets are open sets, but not all open sets are Borel sets.

5. What is the Measure of a Borel Set Minus an Open Set?

The measure of a Borel set minus an open set (denoted as "B minus O") is the difference in size between the two sets. It is the measure of the Borel set that is not covered by the open set. This concept is often used in measure theory to calculate the "missing" portion of a set.

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