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Measured space

  1. Feb 12, 2015 #1


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    Let's say I consider the real numbers and some function real function f, nowhere zero, and positive.
    My question is, what are the conditions on f for dx/f(x) to be a valid measure on this space?

    (I have to consider a Hilbert space [tex] L^2(R, dx/f(x)) [/tex] with scalar product [tex]a.b = \int a^*(x) b(x) \frac{dx}{f(x)} [/tex]

    I'm a physicist, so please excuse me if this is not written in perfect mathematical language!
  2. jcsd
  3. Feb 12, 2015 #2


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    To get a valid measure you just need it (1/f) (to be measurable and) to sum to one; but your Hilbert space will contain only functions a such that the integral of a*a/f is finite. If you need this to apply to a specific class of functions then you'll have to restricf f accordingly.
  4. Feb 12, 2015 #3
    In addition to you assumptions (everywhere or almost everywhere positive) it needs to be measurable, and that is all.
  5. Feb 12, 2015 #4


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    Thanks! And when 1/f is measurable, is it also true for f?
  6. Feb 12, 2015 #5
    Yes, ##1/f## is measurable if and only if ##f## is.
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