Why is the CDF formally defined as $\{\omega \in \Omega : X(\omega) If we have a random variable $X$, then the cdf of $X$ can be formally defined as the events $\{\omega \in \Omega : X(\omega) <x\}$ where $\Omega$ is our sample space. 
My question is, why is it not instead of: 
$\{\omega \in \mathcal{F} : X(\omega) <x\}$?
where $\mathcal{F}$ is the sigma algebra coming from the probability triple $(\Omega, \mathcal{F}, \mathbb{P})$?
 A: Firstly, I think you should ask instead:

Why is the CDF formally defined as $P(\{\omega \in \Omega : X(\omega) <x\})$ instead of $P(\{\omega \in \Omega : X(\omega) \in A\})$ where $A \in \mathcal{B}(\mathbb{R})$?



*

*Recall that $\omega$'s are sample points in the sample space $\Omega$ while the elements of $\mathcal F$ are subsets A, not elements $\omega$, of $\Omega$ s.t. $P(A)$ exists.

*CDF is a probability function computed by the probability measure of a set and not merely a set.


Secondly, it depends on the textbook. All elementary probability texts will define CDF as $P(\{\omega \in \Omega : X(\omega) <x\})$, so why not extend this to advanced probability texts? As for advanced probability texts, some texts will call $P(\{A \in \mathcal{F} : X(\omega) \in A\})$ the law of $X$ denoted by $\mathcal L_X(A)$. The relationship between law of X and cdf of X? By plugging $A = (-\infty, x)$ in the law of X, we get the cdf of X.
Do you know the following?
$$\sigma(\pi(\mathbb R)) = \mathcal B (\mathbb R)$$
where $\pi(\mathbb R) := \{(-\infty,x): x \in \mathbb R\}$
