I am self teaching myself on step functions and measure theory. There was a question I am working on in one of my textbooks. The question is this: Assume $ \{ \phi_n \} $ is a sequence of step functions with $\phi_n \geq 0,n \geq1$, and assume that $\psi_n \uparrow B $ everywhere for a simple function B. Let $P_n = \{x | \phi_n(x) > 0 \}$
Prove that $lim_{n \rightarrow \infty} I(\phi_n) < \infty$ if and only if $lim_{n \rightarrow \infty} \mu^*(P_n) < \infty$
Here is what I have so far. Suppose I assume that $lim_{n \rightarrow \infty} I(\phi_n) < \infty$. Then, I defined a function as follows: $\alpha_n = \phi_n \wedge a_m \chi_{E_m}$. I am not sure I am on the right track here. Can anyone give me some hints and pointers in order to solve this problem?