# Measure of borel set minus open <e

1. Sep 21, 2013

### ArcanaNoir

1. The problem statement, all variables and given/known data
We have a metric space $$X=\cup X_k$$ where $X_k\subset X_{k+1}$ and each $X_k$ is open. Show that for any Borel set E, there is an open set U such that $\mu (U-E)<\epsilon$. (Its supposed to be "U \ E".)

2. Relevant equations

$\mu$ 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.

3. The attempt at a solution

I know that if I have an open set I can intersect it with an $X_k$ 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 $\epsilon$ ?
The only solutions to a problem like this that I have seen are in the context of Lebesgue measure and $\mathbb{R}^n$, 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.

2. Sep 22, 2013

### Jufro

Going back to set notation A/B= A$\cap$ B* (with * denoting the compliment).

This should make it easier to use like like countable subadditivity and finite intersections for measure.

3. Sep 22, 2013

### ArcanaNoir

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. Sep 22, 2013

### pasmith

You're working in a metric space so you should consider looking at a collection (or more than one collection) of open balls.

5. Sep 22, 2013

### Jufro

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= $\cup$some k's Xk.
So E*= ($\cup$k Xk)* = $\cap$ Xk*

and U/E is simply U$\cap$($\cap$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. Sep 22, 2013

### ArcanaNoir

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

Right, I forgot to mention $\mu (X_k)<\infty$

7. Sep 22, 2013

### ArcanaNoir

Yeah, but how would I get only countably many open balls?

8. Sep 22, 2013

### Jufro

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 $\subset$ $\cup$ for some k's Xk.

And then you could use that the fact that if A$\subset$ 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: Sep 22, 2013
9. Sep 23, 2013

### ArcanaNoir

Holy crap I solved it. :surprised