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


In the wikipedia entry of "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 defines the Lebesgue integral of a positive measurable function as \int f d \mu = \text{sup}\{ \int \phi d\mu :\text{    } \phi \text{ is a simple function and } 0 \leq \phi \leq f \} 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?