Clarification on the definition of $\sigma(X)$ with $X$ random variable Let $(\Omega, \mathcal{F}_{| \Omega}, \mu_{| \mathcal{F}})$ be a probability space. Wikipedia states that

The $\sigma$-algebra generated by a random variable X taking values in some measurable space $S$ consists, by definition, of all subsets of $\Omega$ of the form $X^{-1}(U)$, where $U$ is any measurable subset of $S$.

In which sense ''measurable'', i.e., according to which measure?
First of all, since $S$ is a measurable space, it is actually a triple: $S=(\Psi, \mathcal{G}_{| \Psi}, \lambda_{| \mathcal{G}})$ and $\lambda$ is a measure. The subsets $U$ must be measurable according to the measure $\lambda$ ? or according to $\mu$ ? or both? or... ?
Second question: what means, for a set, being a "subset of a measurable space"? Does it mean: $A$ is a subset of $(X,\Sigma,\mu)$ if $A \subseteq X$ and $A$ is $\mu$-measurable?
Last question: Due to the above points, I find a bit difficult to write the definition of $\sigma(X)$ explicitly. Is it something like:
$$
\sigma(X) := \{ E \subseteq \Omega : E = X^{-1}(U), U \in (im(X), \mathcal{G}_{| im(X)}, \lambda_{| \mathcal{G}}) \}
$$
where $im(X)$ is the range of $X$ ?
P.S. Can you suggest me a good textbook regarding measure theory specifically applied to probability?
 A: You seem to be confused as to the difference between a measurable space and a measure space. A measurable space is a pair $S = (\Psi, \mathcal{G})$ where $\Psi$ is a set and $\mathcal{G}$ is a sigma algebra on $\Psi$. A measure space is then a measurable space equipped with a measure on $\mathcal{G}$. Measurability of a function is related only to the measurable space and not the choice of measure on that space.
A measurable subset of a measurable space $S$ is then just an element of $\mathcal{G}$.
So if $X: (\Omega, \mathcal{F}) \to (\Psi,\mathcal{G})$ is a random variable then $$\sigma(X) = \{E \in \mathcal{F}: E = X^{-1}(U), U \in \mathcal{G} \}$$
A: First of all observe that measurability of a function has nothing to do with a measure, i.e. if $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are two measurable spaces, a function $f: X \to Y$ is said to be $\mathcal{A}$-$\mathcal{B}$-measurable if for all $B \in \mathcal{B}$ we have that $f^{-1}(B) \in \mathcal{A}$. In addition, if $(X,\mathcal{A})$ is a measurable space, any set $A \in \mathcal{A}$ is said to be a measurable subset of $X$ with respect to $\mathcal{A}$ for clarification. Also it is easy to write out $\sigma(f)$ explicitely:
$$\sigma(f) = \{f^{-1}(B) : B \in \mathcal{B}\} \subseteq \mathcal{A}$$ In other words:

$\sigma(f)$ is the smallest $\sigma$-algebra which makes $f: X \to Y$
  measurable.

A good textbook which I can recommend is called Probability-1 by Albert N. Shiryaev. It has an excellent introduction to measure theory in the language of probability theory. It is a graduate book, hence the level of formality and abstractness is quite high.
A: 1) If $(\Omega, \mathscr{F})$ is a measurable space, it is customary to briefly say "$\Omega$ is a measurable space". This is not technital at all; you just need to know what to think.
2) Let $(\Omega, \mathscr{F})$, $(S, \mathscr{S})$ be measurable spaces. Then a function $X: \Omega \to S$ is called a random variable if $X$ is measurable, i.e. if $X^{-1}(U) \in \mathscr{F}$ for every $U \in \mathscr{S}$. So the object "random variable" is mathematically just a measurable function. Like a sample space is simply a measurable space. 
3) A subset $E$ of $\Omega$ is called measurable if $E \in \mathscr{F}$. It need not hold that every subset of $\Omega$ is measurable.  
4) We have
$$
\sigma (X) = \{ X^{-1}(U) \mid U \in \mathscr{S} \}.
$$
5) A measure defined on $(\Omega, \mathscr{F})$ is simply a function $\mu: \mathscr{F} \to [0, +\infty]$ such that $\mu (\varnothing) = 0$ and if $E_{1},E_{2},\dots \in \mathscr{F}$ are pairwise disjoint then $\mu (\bigcup_{k}E_{K}) = \sum_{k}\mu(E_{k})$.
6) It seems to me that perhaps you currently lack the very basic concepts or prerequisites of measure theory; so the answers here may not completely solve your doubts in one stroke. You may want to check the books: 
a) K.-L. Chung, A Course in Probability Theory;
b) P. Billingsley, Probability and Measure;
c) A. Zygmund, Measure and Integral.
Book c) can get you in measure theory smoothly without you even realizing it. Book a) can teach you mathematically rigorous probability theory; but you may feel it is kind of scary for now. Book b) may be intuitive and good to many and not so to many too. I personally is not that fond of Book b); however, you are encouraged to take a look at the book to see if his style suits you.
