Lebesgue integral question from wiki

Question

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral.

In the Wikipedia entry on "Lebesgue integration" they define the Lebesgue integral as:

Let $f: \mathbb{R} \rightarrow \mathbb{R}^{+}$ be a positive real-valued function. $$ \int f d\mu = \int_{0}^{\infty}f^{*}(t)dt $$ where $f^{*}(t) = \mu(\{x |f(x) > t\})$.

The Lebesgue integration notes that I am studying define the Lebesgue integral of a positive measurable function as $$ \int f d \mu = \text{sup}\left\{ \int \phi d\mu :\; \phi \text{ is a simple function and } 0 \leq \phi \leq f \right\} $$ 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?

Answer 1

A hint: break the Riemann integral into the limit of a summation, and it should be clear that they are the same. $f^*(t)dt$ is the area on the strip between two step functions that are a distence $dt$ apart.

Answer 2

Define $\int f d\mu$ as usual. (I.e. Supremum of $\sum x \mu(s^{-1}(x))$ where $s$ is a simple measurable function such that $s≦f$.)

Note that $f^*$ is almost everywhere continuous since it is monotonically decreasing. Moreover, it is continuous. (Why?)

So $\int_0^\infty f^*(t)dt$ is well-defined where the integral here is Riemann.

Since $f^*$ is continuous, as Stella mentioned, it can be represented as a summation.

Moreover, do you know a theorem :"For every nonnegative measurable function, there exists a monotonically increasing sequence of simple measurable functions converging to that measurable function"?

Construction in that proof shows that the above summation is exactly the Lebegue integral of $f$.