What does $P_X (A) = P(X^{-1}(A))$ mean in probability? I'm taking a course that covers random variables and I've been trying to make sense of this definition:
Given a set $\Omega$ and a probability $P$, a random variable is a function:
$$X : \Omega \longmapsto \mathbb{R}$$
and the probability of an event $A$ is
$$ P_X (A) = P(X^{-1}(A))$$
The last part is the one that hasn't clicked in yet. To my understanding, a probability is a function $P:\mathfrak{F}\longmapsto [0,1]$, where $\mathfrak{F}$ is a $\sigma$-algebra. I haven't been given a formal definition of $\sigma$-algebra, but I have been told that it's what gives a probability, where $\Omega$ is the universe set, these two properties:

*

*$P(\Omega)=1$

*If $\{A_n\}_{n\geq1}$ is a sequence of disjointed subsets of $\Omega$, then $\bigcup_{i=1}^{+\infty} P(A_i) = \sum_{i=1}^{+\infty} P(A_i)$
So, in the context of the probability of an event $A$, we can say that $A$ is a subset of $\Omega$, but what would $X^{-1}(A)$ mean? Intuitively, if it's something like its counter-image, then its probability would be the probability $P$ of the value of $X$ which corresponds to the event $A$. Is this correct?
I apologize if I wasn't very formal or completely accurate, but this isn't an advanced course and I'm trying to make sense on my own of some things that weren't really explained too thoroughly to us.
Thank you.
 A: $X^{-1}(A)$ means by definition $\{ \omega \in \Omega : X(\omega) \in A \}$. More generally, the preimage (or "inverse image") notation is defined as: if $f : A \to B$ and $C \subset B$ then $f^{-1}(C)=\{ x \in A : f(x) \in C \}$.
Notice that this definition doesn't actually involve the inverse, but it agrees with the ordinary image of the inverse (which would be $\{ f^{-1}(y) : y \in C \}$) if the inverse exists.
Ultimately we can understand $X^{-1}(A)$ informally as just "the event that $X \in A$"; this sort of thing tends to be the better way to think about it since $\Omega$ and its elements are really abstract things.
Another relevant detail that may be confusing you: $P$ and $P_X$ aren't defined on the same domain. $P(A)$ is defined for (some) $A \subset \Omega$; $P_X(A)$ is defined for (some) $A \subset \mathbb{R}$.
A: In this case $A$ is a subset of $\mathbb{R}$ not $\Omega$. The point it that the words "random variable" are "hiding" that your function $X$ is mesurable. So if $A$ is a mesurable set of $\mathbb{R}$ the set $X^{-1}(A)$ has to be in your sigma-algebra, but now you know what the probability of a set in the sigma algebra is because you have a defined probability function.
You are basically trying to "translate" your probability from $\Omega$ to $\mathbb{R}$ via X.
