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Show L^p(E) is separable for any measurable E.

  1. Apr 23, 2013 #1
    I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.
     
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  3. Apr 23, 2013 #2

    micromass

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    You could find an embedding from ##L^p(E)## into ##L^p(\mathbb{R})##.
     
  4. Apr 25, 2013 #3

    Bacle2

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    Don't you mean the simple functions?
     
  5. May 5, 2013 #4
    Though it isn't true if p = infinity, is it?
     
  6. May 10, 2013 #5

    Bacle2

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    You're right, L^oo is not separable.

    Edit: I think there is an argument using simple functions with rational coefficients. It should
    be in most graduate Analysis books.
     
    Last edited: May 10, 2013
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