Suppose there are two random variables $X: \Omega \rightarrow U$ and $Y: \Omega \rightarrow V$ with probability space $(\Omega, \mathcal{F}, P)$, measurable spaces $(U, \mathcal{F}_u)$ and $(V, \mathcal{F}_v)$.
- I was wondering how the joint distribution of $X$ and $Y$ is defined? I just realized I am not clear about its definition.
- If I am correct, the joint distribution is not the same as the product measure of those on $(U, \mathcal{F}_u)$ and $(V, \mathcal{F}_v)$ induced by $X$ and $Y$ from $(\Omega, \mathcal{F}, P)$ respectively. If they are the same, then $X$ and $Y$ are said to be independent.
- Is the joint distribution of $X$ and $Y$ defined on the $\sigma$-algebra generated from $\mathcal{F}_u \times \mathcal{F}_v$?
For $S_u \in \mathcal{F}_u$ and $S_v \in \mathcal{F}_v$, is the joint distribution determined by $P([X, Y] \in S_u \times S_v) = P(\{X \in S_u\} \cap \{Y \in S_v\})$?
How about the joint distribution probability of other sets that are more general and may not be "rectangle"-like as $S_u \times S_v$?
Thanks and regards!