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! :)
On
Well in the proof, since $0\leq f_n\leq f$ we have trivialy that $$ \lim_{n\to \infty} \int f_nd\mu\leq \int fd\mu. $$
So, I like to think like this: I want to proof the oposite inequality. Then
given a simple function $\varphi$, $0\leq \varphi\leq f$ all we want, is to get $f_n$ between $\varphi$ and $f$, i.e,
$0\leq\varphi\le f_n\leq f$. Maybe this could be hard to get
but if we pull $\varphi$ a little bit back :$"\alpha \varphi,~~0< \alpha<1"$ this could be made easier. Well that is the way I like to think about this.
If we won't take $c<1$ we couldn't reach the same result. The point is that $f_n$ increasingly convergence to $f$ and so it's integral becomes larger than the integral of any function that is stricly less than f.
If for example $f=1$ (a constant function, say $X=[0,1]$) and $f_n=1-\frac{1}{n}$ then clearly $f_n$ increasing to $f$. But $f$ is a simple function, hence we can take $s=f=1$. In that case for $c=1$ we have that $E_n = \emptyset$ because $s>f_n$ for every given $n$. Therefore the property that $X=\bigcup E_n$ is not satisfied.