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Showing that a supremum is a maximum

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Let f be a positive integrable continuous function on ℝ. Fix a measurable set E such that E [itex]\subset[/itex] [0,1]. Let

    [itex] s = sup_{β \subset ℝ} [\int_{E} f_{β}(x)dx] [/itex]

    where

    [itex]f_{β}(x) = f(x + β). [/itex]

    Show that s is actually a maximum (not just a supremum) that is, there is at least one β which gives the supremum.

    2. Relevant equations

    (I think they're included in the problem statement...)

    3. The attempt at a solution

    This was an old exam problem from when I took an intro analysis course (Royden) a couple of years ago (2010). I remember I missed something on this exam, but a solution was never posted and the professor never offered one in office hours when I asked. I came across this again and remained puzzled at what exactly had made my proof incorrect... so I'd welcome any thoughts. I know this isn't an attempt per se, but any guidance to start would be great.
     
  2. jcsd
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