Definition of Lebesgue integral as a supremum

Question

Let $X$ be a measure space and $f\colon X\to R$ a Lebesgue-measurable non-negative function. Wikipedia claims that $$\int_X f=\sup_{s\le f} \int_Xs$$ with $s$ running over all step functions bounded by $f$.

Can you provide me with a citable (i.e., non-Wikipedia) reference to this definition and its equivalence to the usual definition?

Answer 1

$\int_X f=\sup_{s\le f} \int_Xs$ is the usual definition for the Lebesgue- integral!

Reference:

D. L. Cohen: Measure Theory, page 63.