I find it very tough to understand Resnick notation for expectation in Probability Path chapter 5 (for which I am not given any kind of explanation nor definition, which is strange), here it is: $$E(X)=\int_\Omega XdP=\int_\Omega X(\omega)P(d\omega)=\int_\mathbb{R}xF(dx) $$
Although I also found some similar questions, I still can't get what actually means' $dP, P(d\omega), F(dx)$', and how they are the same (under differents measure of course). In particular, what is the event $[X\in dx]$ ?
As always, many thanks in advance for any help!
These are all standard notations for the same thing, found in many textbooks. Like all notations, they are conventional. Section 5.2 of Resnick's book is devoted to defining $E(X)$; the first two equations you mention are given as definitions on page 119. The third equation is covered by the contents of section 5.5; see especially p.138.
Your question about the meaning of the event $[X\in dx]$ is more-or-less equivalent to asking "what is the number $dx$?", when confronted with an expression like $\int f(x)dx$ in a calculus textbook. If you are satisfied with an answer like "$\int f(x)dx$ is close to a Riemann sum $\sum f(x_i) (x_i-x_{i-1})$" then you should be satisfied with an answer like "$\int X(\omega)P(d\omega)$ is approximated by a sum of form $\sum_{\Delta_i} X(\omega_i) P(X\in \Delta_i)$, where $\{\Delta_i\}$ is a partition of $\Omega$ for which $\omega_i\in\Delta_i,\forall i$."