Why is the CDF formally defined as $\{\omega \in \Omega : X(\omega)

Question

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 $(\Omega, \mathcal{F}, \mathbb{P})$?

The standard way to do it: $X$ is a function with domain $\Omega$. But subsets of $\Omega$ are not elements of $\Omega$, so $X(\omega)$ makes no sense for $\omega \in \mathcal F$. — GEdgar, May 01 '17 at 02:29
Also, the cdf of $X$ is not the set ${\omega \in \Omega ~:~ X(\omega)< x}$, it is the function $F_X(x)=P({\omega \in \Omega ~:~ X(\omega) \leq x})$, also written $F_X(x)=P(X\leq x)$. (Note the $\leq$ rather than $<$.) — kccu, May 01 '17 at 03:01
Related: Shouldn't a random variable be defined from sets to sets and why the probability density is a distribution? ? — BCLC, Apr 28 '18 at 12:51
Related: Random variables defined on the same probability space with different distributions ? — BCLC, Apr 29 '18 at 08:33

score 1 · Answer 1 · answered Apr 28 '18 at 12:51

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 $\pi(\mathbb R) := \{(-\infty,x): x \in \mathbb R\}$

Why is the CDF formally defined as $\{\omega \in \Omega : X(\omega)

1 Answers1