Monotone Convergence Theorem:

I know this almost follows from the theorem. But I first need to write $\displaystyle \displaystyle \int_{I_n} f = \int_S f_n$ for some $\displaystyle f_n$ in such a way that $\displaystyle (f_n)$ is an increasing sequence tending to $\displaystyle f$. (Then we have something that satisfies the hypotheses of the theorem.) What $\displaystyle f_n$ could I use?

Then in the case of any function $\displaystyle g$ can I consider positive and negative parts?