# Probability space induced by a random variable

Consider three random variables $Y$, $X$, $Z$ all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $Y: \Omega \rightarrow \mathcal{Y} \subseteq \mathbb{R}$, $X: \Omega \rightarrow \mathcal{X} \subseteq \mathbb{R}$, $Z: \Omega \rightarrow \mathcal{Z} \subseteq \mathbb{R}$.

These random variables induce the following probability spaces: $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_Y)$, $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_X)$, $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_Z)$, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$, $P_Y(B):=\mathbb{P}(\{\omega \in \Omega \text{ s.t. }Y(\omega)\in B\})$, $P_X(B):=\mathbb{P}(\{\omega \in \Omega \text{ s.t. }X(\omega)\in B\})$, $P_Z(B):=\mathbb{P}(\{\omega \in \Omega \text{ s.t. }Z(\omega)\in B\})$ for any $B \in \mathcal{B}(\mathbb{R})^k$.

My question is related to the notation: is it correct in terms of notation to write the probability spaces induced by the three random variables with different probability measures $P_Y$, $P_X$, $P_Z$ despite the fact that all the three random variables are defined on the same probability space?

If your answer is :"correct", then I have another question: consider the sequence of real valued random variables $\{X_n\}_n$ and the random variable $X$, all defined on $(\Omega, \mathcal{F}, \mathbb{P})$. The definition with PRECISE notation of the statement "$X_n$ converges in distribution to $X$" is

(a) $\lim_{n \rightarrow \infty} \mathbb{P}(\{\omega \in \Omega \text{ s.t. } X_n(\omega)\leq x\})=\mathbb{P}(\{\omega \in \Omega \text{ s.t. } X(\omega)\leq x\})$ $\forall x \in \mathbb{R}$ where the map $x\rightarrow \mathbb{P}(\{\omega \in \Omega \text{ s.t. } X(\omega)\leq x\})$ is continuous

or

(b) $\lim_{n \rightarrow \infty} P_{X_n}((-\infty, x])=P_X((-\infty, x])$ $\forall x \in \mathbb{R}$ where the map $x\rightarrow P_X((-\infty, x])$ is continuous, $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_{X_n})$ is the probability space induced by $X_n$, $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_X)$ is the probability space induced by $X$.

Correct?

• First question the notation $P_X$ is correct, one rarely needs to write $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_X)$. Second question: (a) is equivalent to the first part of (b) (until "continuous") and both are correct characterizations of the convergence in distribution of $X_n$ to $X$. The rest of (b) is useless. – Did Dec 7 '16 at 13:55
• ((Where in your question do you use X and Y and Z rather than X alone?)) – Did Dec 7 '16 at 13:56