Could you please help me to understand the following concept about the Lebesgue integral from Strichartz's "The way of analysis":
The point is that without monotonicity, or at least some such restriction, there need be no relationship between the integral of $f$ and the limit of the integrals of $f_n$, where $f_n$ are simple functions such that $\lim f_n(x)=f(x)$ pointwise.