Clarifications of measure theoretic definition of a random variable

Question

Wikipedia defines a random variable:

Let $(\Omega,F,P)$ be a probability space and $(E,S)$ a measurable space. Then an $(E,S)$-valued random variable is a measurable function $X:\Omega \to E$, which means that, for every subset $B \in S$, its pre-image $X^{-1}(B)\in F$ where $X^{-1}(B)= \{\omega: X(\omega)\in B\}$. This definition enables us to measure any subset $B \in S$ in the target space by looking at its pre-image, which by assumption is measurable.

Question 1. Do all $E$-values in $E$ to which the random variable maps have to be included as part of some event in $S$? Since $S$ is a sigma-algebra in measurable space $(E,S)$ it has to include all of $E$, right?

Question 2. One of the elements in $S$ will always be $E$ (since $S$ is a sigma-algebra on $E$) and the same goes for $(\Omega,F)$. Does the random variable always map entire $E$ to probability of entire $\Omega$ (i.e. probability of entire $E$ is always 1)?

Question 3. Is it possible to have events in $S$ that do not have probability i.e. are not mapped to any event in $F$?

I realise the questions overlap somewhat.

Answer 1

You are getting confused between image and inverse image.

$E$ is certainly in $S$ and the inverse image of $E$ under any map from $\Omega $ to $E$ is always $\Omega$.

For every set $F$ is $S$ certainly $P(X \in F)=P(X^{-1}(F))$ is well defined.

Answer 2

1) This question is unclear because there is no common notion of "$ E$-valued elements of $E$".

Further $E\in S$ because $S$ is a $\sigma$-algebra and of course $E$ contains every element of $E$. This is actually a tautology.

2) $X^{-1}(E):=\{\omega\in\Omega\mid X(\omega)\in E\}$ and evidently every element of $\Omega$ is sent to $E$. So we conclude that $X^{-1}(E)=\Omega$ and consequently: $$P_X(E)=P(X^{-1}(E))=P(\Omega)=1$$

3) $X$ is not mapping elements of $S$ to events of $F$. It maps elements of $\Omega$ to elements of $E$. Then for every element $B$ of $S$ there is a preimage under $X$ which is a subset of $\Omega$. The measurability of $X$ ensures that all these preimages are elements of $F$ so that they can all be measured by $P$. That opens the door for defining a probability measure $P_X$ on $(E,S)$ by stating that: $$P_X(B):=P(X^{-1}(B))$$