- #1

ArcanaNoir

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## 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.