Why is the convergence only a.e.? ($L^p$ spaces)

Question

In this proposition for $L^p$ spaces:

Why is the convergence only a.e.? If I divide the function in real and imaginary parts, and then in positive and negative parts for both of these. I get that I can use the fact that for every non-negative measurable function, I can get a sequence of simple functions converging to that sequence pointwise. And here, the convergence is everywhere? So why is it written a.e.?

Answer 1

Because elements of $\mathcal L^p$ are not functions, but equivalence classes of functions up to measure $0$. Hence for any specific representavie of the same class we cannot say anything better than "almost everywhere".