Normally, Lebesgue integral, for positive measures, is defined in the following way. First, one defines the integral for indicator functions, and linearly extend to simple functions. Then, for a general non-negative function $f$ , it is defined as the supremum of the integral of all simple functions less than or equal $f$.
However, I find another definition in the book of Lieb and Loss, "Analysis". Let $f$ be the non-negative measurable function on a measure space $X$, let $\mu$ be the measure, and define for $t >0$, $$S_f(t) = \{x \in X : f(x) >t\},$$ and $$F_f(t) = \mu(S_f(t)) .$$
Note that $F_f(t)$ is now a Riemann integrable function. Now, the Lebesgue integral is defined as:
$$ \int_X f \, d\mu = \int_0^\infty F_f(t) \, dt,$$
where the integral on the right-hand side is the Riemann integral.
Here is the google books link to the definition.
The book gives a heuristic reason why this definition agrees with the usual definition described here in the first paragraph. Now I would like to have a rigorous proof of the equivalence.