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I have started studying Lebesgue integration and I have a few of questions regarding the Lebesgue integral:

  1. In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: $\int f d\mu = \int_{0}^{\infty}f^{*}(t)dt$ where $f^{*}(t) = \mu(\{x |f(x) > t\})$. The Lebesgue integration notes that I am studying first defines the integral of the positive simple function on a measure space, then the positive measurable function followed by the sign changing measurable function. I want to know if this wiki definition is equivalent to the integral constructed from simple functions, if so how can this be easily shown?

  2. Secondly , I want to know how the Lebesgue integral is used exactly. I understand how it is defined in terms of simple functions, in the same way I know how Riemann integration is defined by the limit of the Riemann sum. But I also want to know if there are equivalent theorems and techniques for Lebesgue integration as in Riemann integration used when actually computing a given integral. For example how does the following translate to Lebesgue integration: the evaluation theorem, using anti-derivatives to evaluate indefinite integrals, the fundamental theorem of calculus, the substitution rule and integration by parts?

  3. Lastly as an example for the integral $\int x^{2} dx$ evaluated as a Lebesgue integral, would you evaluate it as you would as a Riemann integral by taking the anti derivative? Is this always the case?

Thanks a lot for any assistance.

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The main practical use of the Lebesgue integral is for knowing that it exists. Since it is easier for a function to be Lebesgue integrable than Riemann integrable, there are more theorems of the form "If such-and-such, then this-or-that function is Lebesgue integrable" than there would be for Riemann integrals.

On the other hand, it is a theorem that IF a function is Riemann integrable, then it is Lebesgue integrable with the same integral. So in practice if you have a concrete function given in a nice closed form and want to know its Lebesgue integral, what you actually do is try to find its Riemann integral using antiderivatives (and by extension the ordinary toolkit for finding antiderivatives -- substitution, integration by parts and so on). If you succeed in Riemann-integrating it, what you get is guaranteed to be the Lebesgue integral too.

Cases where you need to compute the Lebesgue integral of something that is not Riemann integrable (and cannot be handled as improper integrals) are rare in practice, and there is no standard way to handle them when they do arise.

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  • $\begingroup$ Thanks for your response. You say that "it is easier for a function to be Lebesgue integrable than Riemann integrable", why is this the case? Also what is the great significance of Lebesgue integration if you very rarely encounter a integral which is not Riemann integrable? Would I be right in stating that the significance of Lebesgue integration in practice is in using the Lebesgue integration theorems such as Dominated Convergence Theorem, Beppo Levi and Fatous's Lemma to calculate Riemann integrals? $\endgroup$ – user103184 Feb 28 '14 at 11:24

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