Showing that a particular G_delta set exists with a measure property

[jdinatale]

Ok, I don't think I'm on the right track here. I ASSUMED that the set of all countable collections $\{I_k\}_{k = 1}^\infty$ of nonempty open, bounded intervals such that $E \subseteq \bigcup_{k = 1}^\infty I_k$ is a countable set itself, which it probably isn't.

I'm not even sure where to start on this problem. I feel like I need to use the assumption that E is bounded. I know if E is bounded, then E can be covered by a finite number of nonempty open, bounded intervals.

 

[gopher_p]

Ok, I don't think I'm on the right track here. I ASSUMED that the set of all countable collections $\{I_k\}_{k = 1}^\infty$ of nonempty open, bounded intervals such that $E \subseteq \bigcup_{k = 1}^\infty I_k$ is a countable set itself, which it probably isn't.

It's not. If you had a countable family of covers of this type, could you show that their intersection was a $G_\delta$ set? Presumably that's the direction that you were going with this part of the argument. Then you'd at least have a $G_\delta$ cover, and all that is left is to rig it so that it has the correct measure.

I'm not even sure where to start on this problem. I feel like I need to use the assumption that E is bounded. I know if E is bounded, then E can be covered by a finite number of nonempty open, bounded intervals.

You have a decent start. You just don't have any control (measure-wise) over your covers. You need to use the assumption that $E$ is bounded to get that control. If $E$ is bounded, what can you say about its outer measure?