What is the type (?) of this term: $X \leq x$ in the definition of the CDF?
$F_{X}(x) = P(X \leq x)$
Is $X \leq x$ a set? Is it real valued? I know that ${P}$ is a function that maps to $[0,1]$. Does it always map a set to $[0,1]$? How would I read this aloud?
The notation $ P (X \le x) $ simply says "the probability that a random variable $ X $ is less than or equal to a value $ x $ in its event space."
So, assuming the order relation $\le $ makes sense on the event space, then $ X \le x $ can be a set, just like $ X \le 4$ is a subset of the reals.
$P(X\le x)$ is read as
"the probability of the event $(X\le x)$".
In the abstract way, a set $\Omega$ of 'elementary events' is given, and certain subsets of it are declared as 'events', and actually these are just the measurable subsets for the probability measure $P$ that assigns each event a probability.
I like to think of elements of the abstract $\Omega$ as the 'potential continuations of the given random experiment', though in the calculations this doesn't matter much.
Nonetheless, formally the event $(X\le x)$ is just the set $$(X\le x)\ :=\ \{\omega\in\Omega\ \mid\ X(\omega)\le x\}$$ where $X:\Omega\to\Bbb R$ is a random variable, i.e. a measurable function from the probability space $\Omega$, and that basically means that all the sets of the above type (or similarly formulated) are actually events, i.e. $P$ is defined for them.
