Definition of a real-valued random variable I have trouble understanding the definition of a random variable:

Let $(\Omega, \cal B, P )$ be a probability space. Let $( \mathbb{R}, \cal R)$ be the usual
  measurable space of reals and its Borel $\sigma$- algebra. A random variable
  is a function $X : \Omega \rightarrow \mathbb{R}$ such that the preimage of any set $A \in \cal R$ is
  measurable in $\cal B$: $X^{-1}(A) = \{ w: X(w) \in A \} \in \cal B$. This allows us to define
  the following (the first P is the new definition, while the 2nd and
  3rd Ps are the already-defined probability measure on $ \cal B$):
$ P(X \in A) = P(X^{-1}(A)) = P(\{ w : X(w) \in A \}) $
$ P(X = x ) = P(X^{-1}(x)) = P(\{ w : X(w) = x \}) $

Does it mean that the first P (the leftmost one) is a new probability measure defined on $( \mathbb{R}, \cal R)$? (in contrast with the 2nd and 3rd Ps defined on $(\Omega, \cal B)$)
What is $(\Omega, \cal B, P )$ for a binomial distribution for example?
In which cases do 2 random variables share the same probability space $(\Omega, \cal B, P )$?
 A: The probability of an event $X \in A$ is, by definition, the $P$-measure of the set of "outcomes" $\omega$ for which $X(\omega)$ is in $A$.  Strictly speaking, all events are measurable subsets of the sample space, but it's usually simpler to speak of
events involving  random variables without explicitly mentioning the sample space.
For example, if your random variable $X$ is the number of successes in 
$n$ independent Bernoulli trials with probability of success $p$ in each one
(and thus has the binomial($n$,$p$) distribution), the sample space $\Omega$
could be $\{0,1\}^n$ (where $0$ corresponds to failure and $1$ to success).  $\mathcal B$ would be all subsets of $\Omega$, and 
$P$ gives each outcome with $k$ 1's and $n-k$ 0's probability $p^k (1-p)^{n-k}$.
But then if you want to relate this to some other random variable $Y$ that is
not determined by those same Bernoulli trials, you'll need a bigger sample space, where each outcome consists not just of the outcomes of those $n$ trials but also something else that determines $Y$.
EDIT: This is what distinguishes probabilists from real analysts.  The
analyst is studying  real-valued functions on a given space $\Omega$ with
a given $\sigma$-algebra $\mathcal B$ and probability measure $P$.
The probabilist will use the rigourous definition of the random variable $X$ in terms of $\Omega$ and $\mathcal B$ if necessary, but really thinks of $X$ in terms of a quantity involved in some (actual or imagined) experiment, and he/she is willing to change $\Omega$ and $\mathcal B$ in midsentence if that becomes convenient. 
