Riemann-Sieltjes vs. Lebesgue

jsr9119

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

micromass

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.

Stephen Tashi

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.)

mathman

Lebesgue-Stieljes integral is best described as Lebesgue integration with respect to a given measure.

pwsnafu

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.

micromass

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.