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Showing that a particular G_delta set exists with a measure property

  1. Aug 26, 2013 #1
    png_zps63374024.png

    Ok, I don't think I'm on the right track here. I ASSUMED that the set of all countable collections [itex]\{I_k\}_{k = 1}^\infty[/itex] of nonempty open, bounded intervals such that [itex]E \subseteq \bigcup_{k = 1}^\infty I_k[/itex] 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.
     
  2. jcsd
  3. Aug 26, 2013 #2
    It's not. If you had a countable family of covers of this type, could you show that their intersection was a [itex]G_\delta[/itex] set? Presumably that's the direction that you were going with this part of the argument. Then you'd at least have a [itex]G_\delta[/itex] cover, and all that is left is to rig it so that it has the correct measure.

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