Let $\Omega = \{-2,-1,1,2\}$ and $\mathbb P :\mathscr P(\Omega)\to[0,1]$such that: $$ \mathbb P (F)=\frac{\# F}{\# \Omega}=\frac{\# F}{4} $$ and let $X$ be a random variable such that $X(\omega)=1+\omega^2$
Now, let $\mathbb Q :=\mathbb P^X : \mathscr B (\mathbb R)\to[0,1]$ be the distribution of X under $\mathbb P$.
I am really confused about $\mathbb Q$, how is this measure defined? How does one begin investigating it?
$X$ is a discrete random variable with $$P\{X = 2\} = P\{X= 5\} = 1/2.$$
Because this $\Omega$ is finite, it seems to me we are making the usual choice that the sigma-algebra events, on which probabilities can be defined, consists of all subsets of $\Omega$ (the power class), and that the measure is 'counting measure' of points. (Notation is not exactly standard, so some words might have helped.)