Quote by Haelfix
Look on page 6, they use absolute convergence (not absolute continuity, I never said anything about that) in the definition of what is or is not integrable.

They use absolute convergence when refering to the integrability of a simple function. Which makes some sense, since the Lebesgue integral of a simple function is defined with a summation, and you want that summation to converge absolutely.
But saying that integrability requires absolute convergence makes no sense when refering to an arbitrary function. Absolute convergence is a property of a series, not of a function.