# Understanding Lebesgue Integration

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.