Definition of Random Variable on Measure Theory! The definition is as following according to the book of John B. Walsh,

Let $(\Omega, \mathbb{F}, P)$ be a probability space. A Random Variable is a real-valued function X on $\Omega$ such that for all $x \in \mathbb{R}, \{\omega:X(\omega) \leq x\} \in \mathbb{F}$

my simple question is why the definition limits for each event of R.V. to be less than or equal a real number x? (I just want to know the intuitive of the $X(\omega) \leq x$).
In the book he argues that when you want to prove say X + Y (where X and Y both are R.V.) first we prove that $\{\omega : X(\omega) + Y(\omega) > x\} = \bigcup_{r \in \mathbb{Q}}\{\omega : X(\omega) > r, Y(\omega) > x - r\}$. Then verify that this is in F, and conclude that $x \in \mathbb{R}, {\omega:X(\omega) + Y(\omega) \leq x} \in \mathbb{F}$. Why he could go to prove that the R.V. of X and Y is greater than a real number x, then he concludes that they are less than or equal to a real number x? (This make me confuse about what is behind the definition of R.V.)
R.V. : Random Variable.
 A: The random variable is not limited to values less than $x$. For instance, I can show you that the function 
$$
X(t) = \frac{1}{t}
$$
is a measurable function on $(0, 1)$. Here's how. Let's look at 
$$
\{ t \mid X(t) \le 11 \}
$$
That's the set of all points in the domain for which $X(t) = 1/t$ is less than 11, which is exactly
$$
A = \{t \mid \frac{1}{11} \le t < 1 \}
$$
Clear enough? Now look at $A$. Is it a measurable set in the measure space $(0, 1)$? Sure. It's a half-open interval, and those are all measurable sets!
Now there was nothing special about the number 11: I could have chosen $6$ or $113$ or $-12$ (although if I'd chosen $-12$, then the set $A$ woudl have been empty. Fortunately, the empty set is measurable, too!
Now go back and look at the definition above: it says "whenever you build a set like $A$, it's measurable." It does not say that the function $X$ is bounded. Indeed, in my case, we have
$$
X(0.1) = 10 \\
X(0.01) = 100 \\
X(0.001) = 1000
$$
and so on, and it's pretty clear that $X$ itself is an unbounded function on $(0, 1)$. 
A: The actual definition is that a function $X:\Omega\to\mathbb R$ is called a random variable when $X$ is $(\mathcal F, \mathcal B(\mathbb R))$-measurable. In other words, $X^{-1}(B)\in\mathcal F$ for any $B\in\mathcal B(\mathbb R)$, where $\mathcal B(\mathbb R)$ is the Borel $\sigma$-algebra on $\mathbb R$.
Now, it is sufficient that $X^{-1}((-\infty,x])\in\mathcal F$ for each $x\in\mathbb R$, as
$$\mathcal B(\mathbb R) = \sigma\left(\{(-\infty, x] : x\in\mathbb R\}\right) $$
(this is a good exercise; it follows readily from the definition of the Borel sets as the $\sigma$-algebra generated by the open sets in $\mathbb R$, and that the collection of open intervals $\{(a,b) : a<b\}$ is a basis for this topology).  To see this, let $a<b$, then $(a,\infty)=(-\infty,a]^c$ and
$$(a,b) = (a,\infty)\cap\bigcup_{q\in\mathbb Q,\ q<b}(-\infty, q], $$
so $\sigma\left(\{(-\infty, x] : x\in\mathbb R\}\right)$ is a $\sigma$-algebra containing the open intervals, and similarly 
$$(-\infty, x] = \bigcup_{n=1}^\infty (x-n,x) \cup \bigcap_{n=1}^\infty \left(x-\frac1n, x+\frac1n\right), $$
so $\mathcal B(\mathbb R)$ is a $\sigma$-algebra containing the sets $(-\infty,x]$.
The reasoning for using sets of the form $(-\infty,x]$ is that we define the distribution function of a random variable $X$ by
$$F(x) = \mathbb P(\{\omega\in\Omega : X(\omega) \leqslant x\}), $$
or $\mathbb P(X\leqslant x)$ for short.
A: A random variable $X$ on $\Omega$ is no more and no less than a function $X:\>\Omega\to{\mathbb R}$ satisfying the technical condition that it is measurable: For any $x\in{\mathbb R}$ the set $\{\omega\in\Omega\>|\>X(\omega)\leq x\}$ belongs to ${\cal F}$. This guarantees that for any two given values $a$, $b$ the probability $$P[a\leq X(\omega)\leq b]=\int_\Omega 1[a\leq X(\omega)\leq b]\>{\rm d}\mu(\omega)$$ is well defined. Note that there is nothing "random" in this definition.
What is random is the following: Fate selects the point $\omega\in\Omega$ where the function $X$ is evaluated.
