"Let $(\Omega, \mathcal{F}, \mathcal{P})$ be a probability space..." is a typical phrase found in scientific publications. There are a couple of questions regarding this notation.
First, about the probability measure $\mathcal{P}: \Omega \to [0, 1]$. My confusion comes from the fact that, after such an introduction, authors typically start operating on different random variables, which have some particular distribution functions, say $X_i(\omega) \sim \Phi_{X_i}(x)$. $\Phi_{X_i}(x)$, $\forall i$, seem (to me) to be related to different probability measures, which cannot be covered by a signle one, $\mathcal{P}$. Therefore, for each $X_i(\omega)$ I would expect one to define a separate probability space $(\Omega, \mathcal{F}, \mathcal{P}_i)$ such that $F_{X_i}(x) = \mathcal{P}_i(X_i(\omega) \leq x)$. However, I do not see anything like this. There is, probabily, some misunderstanding from my side.
Second, about the corresponding space of square-integrable random variables. Usually, it is denoted by $L^2(\dots)$ with different variations of what is in the brackets. I suppose "square-integrable" makes little sense without a measure; therefore, it should be at least $L^2(\Omega, \mathcal{P})$, but quite often one can find just $L^2(\Omega)$.
Can anyone please comment on this? Thank you.
Regards, Ivan