If $(\Omega ,\mathcal F,\mathbb P)$ is a measurable set and $X:\Omega \to R $ measurable, what is concretely $(\Omega ,\mathcal F, \mathbb P_X,)$?

Question

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. Let $$X:\Omega \longrightarrow \mathbb R$$ a random variable. In particular, we can define on $\Omega $ the probability :

$$\mathbb P_X(A)=\int_A X\mathrm d \mathbb P=\mathbb P\{X\in A\}.$$

1) Why is the connexion between $(\Omega ,\mathbb F, \mathbb P)$ and $(\Omega ,\mathcal F,\mathbb P_X)$. I guess that to describe $X$ we must be in $(\Omega ,\mathcal F,\mathbb P_X)$, no ? I'm very confuse with those two space. Someone could explain clearly ?

2) Moreover, in the previous question, where is $A$ ? Is it a Borel set ? Or may be $A$ live in an other $\sigma -$algebra ?

3) By the way, a priori $\sigma (X)\subsetneq\mathcal F$, so shouldn't it be $(\Omega ,\sigma (X),\mathbb P_X)$ instead of $(\Omega ,\mathcal F,\mathbb P_X)$ ?

Answer 1

If $(\Omega,\mathcal F,\mathbb P)$ denotes a probability space then $X$ is a random variable defined on this space if it is a function $\Omega\to\mathbb R$ that is measurable in the sense that for every Borel set $A\subseteq\mathbb R$ we have: $$\{X\in A\}=\{\omega\in\Omega\mid X(\omega)\in A\}\in\mathcal F$$

Random variable $X$ then induces a probability measure on measurable space $(\mathbb R,\mathcal B)$ where $\mathcal B$ denotes the $\sigma$-algebra of Borel subsets of $\mathbb R$. A notation for that probability measure is $\mathbb P_X$, and it is prescribed by:$$A\mapsto\mathbb P(\{X\in A\})$$

Note that the outcome space for $\mathbb P_X$ is $\mathbb R$ and not $\Omega$ as you wrongly suggested. This probability measure is what we call the distribution of $X$.

It happens that random variables $X$ and $Y$ are incomparable since they are not defined on the same probability space. However comparable distributions $\mathbb P_X$ and $\mathbb P_Y$ are induced. This is used intensively and somehow reveals the importance of this all.

1) We have the original space $(\Omega,\mathcal F,\mathbb P)$ and for every random variable $X$ on it we have the induced space $(\mathbb R,\mathcal B,\mathbb P_X)$ with the link $\mathbb P_X(A)=\mathbb P(\{X\in A\})$ for every $B\in\mathcal B$. This link can be extended to integrals:$$\int f(X(\omega))\mathbb P(d\omega)=\int f(x)\mathbb P_X(dx)$$where $f:\mathbb R\to\mathbb R$ denotes an integrable function on space $(\mathbb R,\mathcal B,\mathbb P_X)$.

2) Yes, $A$ denotes a Borel subset of $\mathbb R$. I restrict in my answer to random variables $\Omega\to\mathbb R$.

3) if $\sigma(X)$ is not a subset of $\mathcal F$ then $X$ is not a random variable on $(\Omega,\mathcal F,\mathbb P)$. In that case there are Borelsets $A\subseteq\mathbb R$ such that $\{X\in A\}\notin\mathcal F$ and consequently there is no definition for the probability that $X\in A$ "happens".

Answer 2

In this notation you provide $\mathbb R$ with the Borel sigma algebra and start with the assumption that $X$ is a random variable on $(\Omega, \mathcal F, P)$, i..e $X^{-1}(A) \in \mathcal F$ for all Borel sets $A$. Under this condition $P_X (A)$ makes sense for any Borel set $A$. $P_X$ is a measure on the Borel sigma algebra of $\mathbb R$ and it carries full information about the distribution of the random variable, but not all information about $X$. As far as 3) is concerned note that you don't have to restrict to $\sigma(X)$ as along as $X^{-1}(A) \in \mathcal F$ for all Borel sets $A$. By default $\mathbb R$ is given the Borel sigma algebra when you are dealing with real valued random variables, so no explicit mention of Borel sigma algebra has been made in the definiton of $P_X$. The importance of $P_X$ comes from the fact that $P\{X\leq x\}=P_X(-\infty, x]$ for all $x \in \mathbb R$.