# Measure of borel set minus open <e

## Homework Statement

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".)

## Homework 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.

## 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.

## Answers and Replies

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.

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.

pasmith
Homework Helper
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.

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.

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.

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 $\mu (X_k)<\infty$

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?

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.

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Holy crap I solved it. :surprised