Could someone explain me the meaning of the following expected value of a positive random variables $X$?
$\mathbb E[X] = \sup(\{\mathbb E[Y] : Y\text{ a simple r.v. with }0 < Y < X\})$
where simple random variable means: $$Y=\sum_{i=1}^n a_i\mathbf1_{A_i}$$
On
For a nonnegative simple random variable $Y=\sum_{i=1}^na_i\mathbf1_{A_i}$ we have a definition of its expectation like this:$$\mathbb EY:=\sum_{i=1}^na_iP(A_i)\tag1$$What we want is a more general definition for nonnegative random variables and this can be achieved by stating that - if $X$ is a nonnegative random variable:$$\mathbb EX:=\sup(\{\mathbb EY\mid Y\text{ is a simple nonnegative random variable that satisfies }Y\leq X\})\tag2$$
This gives a definition for a greater class of random variables.
Observe however that for nonnegative simple random variable $Y$ we now have $2$ definitions. It must be checked that $\mathbb EY$ according to $(1)$ is the same as $\mathbb EY$ according to $(2)$ (where $X$ is replaced by $Y$ and $Y$ by e.g. $Z$), and fortunately that is indeed the case.
It is a good exercise to this check yourself.
Define $Y_n=\frac {k-1} {2^{n}}$ when $\frac {k-1} {2^{n}} \leq X <\frac k {2^{n}}$ for $1\leq k <n2^{n}$ and $Y_n=n$ when $X >n$. Then $Y_n$ is a simple function, $0\leq Y_n \leq X$ and $Y_n$ increases to $X$ as $n$ increases to $\infty$. By Monotone Convergence Theorem $EY_n \to X$. It follows that $\sup \{EY:0\leq Y\leq X, Y \text {simple}\} \geq EX$. The reverse inequality is obvious.