so, a finite measure characterize this space

by the way, the first question is also in Big Rudin.

You are right. BTW, is there any relationship there? Possion, geometric, gamma are related by conditioning this distribution. I am impressed by this wonderful relationship.

Thanks, transition kernel or probability kernel are introduced in this book, but i feel difficult when evaluating some conditional probability/expectation. To be precise, I don't know how to find a starting point, like should I first calculate the conditional expectation? or should I just use so called conditional density, etc.

ok, you are right, I check the table of notation, Z+ includes zero. But how to compute the law of X knowing Y?

but X is r.v. takes values in the postive integer