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Riemann-Sieltjes vs. Lebesgue

  1. Dec 4, 2012 #1
    Hey guys,

    I'm doing a paper on the Radon transform and several sources I've come across cite the Lebesgue integral as a necessary tool to handle measures in higher order transforms.
    But, Radon's original paper employs the Riemann-Stieltjes integral in its place.

    I read that Lebesgue is more general and so Radon could have used it in place of RSI. Is this the case?

    Thanks,

    Jeff
     
  2. jcsd
  3. Dec 5, 2012 #2
    The Lebesgue integral is indeed more general than the Riemann integral.
    Using measure theory, we can also develop the Lebesgue-Stieltjes integral, and this is a generalization of the Riemann-Stieltjes integral.

    So yes, the paper could probably be written with Lebesgue instead of Riemann. But there may be technical differences between the two.
     
  4. Dec 5, 2012 #3

    Stephen Tashi

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    Is there a distinction between "Lebesgue Integration" and "integration with respect to Lebesgue measure"? My impressions is that "Lebesgue measure" on the real number line is a particular measure that implements the usual notion of length, so the measure of a single point would be zero. On the other hand, is "Lebesgue integration" defined with respect to an arbitrary measure?

    For Lebesgue Integration to include Riemann-Stieljes integration as a special case, is it necessary to use measures other than Legesgue measure? (I'm thinking of the specific example of defining an integration that can integrate a discrete probability density function by the method of assigning non-zero measure to certain isolated points and turning "integration" into summation.)
     
  5. Dec 5, 2012 #4

    mathman

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    Lebesgue-Stieljes integral is best described as Lebesgue integration with respect to a given measure.
     
  6. Dec 5, 2012 #5

    pwsnafu

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    Yes. Lebesgue integration is defined with respect to a measure. The procedure is the same, but different measures give different integrals.

    The Lebesgue measure is derived from the set function ##m((a,b])=b-a##.
    The Stieljes measure is derived from the set function ##m(a,b]) = g(b)-g(a)## for some monotonically increasing function g.

    Summation is a special case of Lebesgue integration, using the counting measure over Z, or Dirac measure over R.
     
  7. Dec 5, 2012 #6
    In my experience, it's a bit ambiguous. When talking about Lebesgue integration, sometimes people talk about general integration wrt a measure and sometimes they talk about integration wrt Lebesgue measure. It's usually clear from the context though.
     
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