Probability and random variables notation

Question

I have a probability space $(\Omega, S, P)$, where $\Omega$ is a sample space, $S$ is a sigma-algebra, $P$ is probability measure, $P : S \to [0,1]$. Now I consider some other space $(A, B)$, where $B$ is a sigma-algebra on $A$. This allows me to define a random variable as a measurable function $X : \Omega \to B$ such that for every $\mathcal{B} \in B$ I have $X^{-1} (\mathcal{B}) = \{ \omega \in \Omega : X(\omega) \in \mathcal{B} \} \in S$. Finally I can bring in the probability distribution $\mathbb{P}$ which is another measure, this time defined as $\mathbb{P} : B \to [0,1]$, such that for any $\mathcal{B} \in B$ we have $\mathbb{P} (\mathcal{B}) = P(X^{-1} (\mathcal{B}))$. This is, I believe, a pushforward measure, literally from the definition, correct?
So now I have some question regarding notation, for example in some textbooks I may encounter random variable $X \sim N(0,1)$ and some exercise to calculate $P ( -1 < X <2 )$. My question is what is $P$ in this case? Is it my original $P : S \to [0,1]$, or is it what I've defined as $\mathbb{P}$? Or maybe is it some not-exactly-formal thing to make the notation easier on the eyes?

Answer 1

In $P(-1<X<2)$ $P$ stands for the original probability measure on $\Omega$. In terms of $\mathbb P$ we can write $P(-1<X<2)=\mathbb P ((-1,2))$ since $X^{-1}((-1,2))=\{\omega \in \Omega: X(\omega) \in (-1,2)\}=\{\omega: -1<X(\omega) <2\}$.