The proof is the following
The proposition 1.25 and Theorem 1.19(d) that refers to are these
Why do we need that constant $0 <c <1$ in the proof of Leb. Mon. Conv. Thm? For me, the proof works fine if we take just any simple function $s $ and define the sets $E_n $ with $s $ instead of $cs$.
Thank you! :)