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Integrable function proof

  1. May 10, 2006 #1
    A is the open unit ball in R^n. Let B be the compliment of A (R^n\A).
    If f: B -> R is defined by f(x) = ||x||^-3... (where x is in B)

    For n=2, using an increasing union of compact sets show that f is integrable on B.

    For n=3, show that f is not integrable.

    Does an increasing union of sets here mean that each compact set must be contained entirely in the next? It seems clear here that f will be bounded for n=2 (from 0 to 1), and thus would suggest that it is integrable, but then why not n=3? I seem to be missing a requirement for f being integrable here, any help would be appreciated.
  2. jcsd
  3. May 10, 2006 #2
    I did this question (I'm assuming that you've just taken this from the assignment) by choosing an obvious increasing union of compact sets. Polar coordinates, spherical coordinates hmm...

    The fact that f is bounded doesn't really tell you that much. If you look at the definition you need a certain limit to exist.
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