Can we define a joint probability distribution over different sample spaces / probability spaces? I am trying to understand the concept of a joint probability distribution. According to the definition of a (say) bivariate probability distribution, the two random variables $X$ and $Y$ should be defined on the same probability space.
http://en.wikipedia.org/wiki/Joint_probability_distribution

Are there such things as joint probability distributions over different sample spaces or different probability space?
Of course you can. The joint probability has Cartesian product of each own's sample space as joint sample space and product $\sigma-$algebra as joint events space. The joint probability, obviously, is the product of probabilities of $X$ and $Y$.
Of course you can. The joint probability has Cartesian product of each own's sample space as joint sample space and product $\sigma-$algebra as joint events space. The joint probability, obviously, is the product of probabilities of $X$ and $Y$.