In 2.4.2 of Geometric Measure Theory, Federer gives the following description of the upper (Lebesgue) integral. Given a (not necessarily measurable) function $f : X \to \mathbb{R}$ on a measure space $(X, \mathcal{A}, \mu)$, we call $u : X \to \mathbb{R}$ an upper function of $f$ if $f \leq u$ almost everywhere and $u$ is a measurable function with countable range. We then define the upper integral of $f$ to be the infimum $$\int^* f \mathrm{d} \mu : = \inf \left\{ \int u \mathrm{d} \mu : \textrm{$u$ is an upper function for $f$} \right\} .$$ Lower functions and lower integrals are then defined analogously. Federer then says that a function is integrable if it's measurable and its upper and lower integrals agree, and proceeds with actual measure theory.
My question is this. Do the classical integral convergence theorems like Monotone Convergence and Dominated Convergence hold for the upper integral? If not, is there an easy counterexample? And is there anywhere I can read more about these upper integrals?
Thanks!