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"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.

  1. 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.

  2. 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

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1 Answer 1

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  1. Actually $\mathcal{P}$ is a map $\mathcal{F} \to [0,1]$, not $\Omega \to [0,1]$, and if $X : \Omega \to \mathbb{R}$ is a random variable then $\mathcal{P}(X \le x)$ is actually an 'abuse of notation' used to mean $$\mathcal{P}(X \le x) := \mathcal{P}( \left\{ \omega \in \Omega\, :\, X(\omega) \le x \right\} ) = \mathcal{P} \circ X^{-1}((-\infty, x])$$ and so $\mathcal{P}$ does not depend on the distribution of $X$. [Notice that $(-\infty, x]$ is a Borel-measurable subset of $\mathbb{R}$, so $X^{-1}((-\infty, x])$ is $\mathcal{P}$-measurable (i.e. it lies in $\mathcal{F}$), and so the expression makes sense.]

  2. Square-integrable functions are defined over a probability (or measure) space, so in fact it is $L^2(\Omega, \mathcal{F}, \mathcal{P})$, though sometimes $L^2(\Omega)$ or $L^2(\mathcal{P})$, or even $L^2$, are used as shorthands when the other parameters are understood from context.

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