Assume we have an algorithm which asks random oracle $\mathcal{O}$ $Q$ queries $u_1, \ldots, u_Q$. All queries are unique, $u_i \neq u_j$ for $i \neq j$. Queries $u_i$ are random variables, too.
What is the easiest (mathematically correct) way to see that the random variable $(\mathcal{O}(u_1),\ldots,\mathcal{O}(u_Q))$ has uniform distribution -- or -- that $\mathcal{O}(u_i)$ are independent random variables?
It looks like that the standard theorems covered in textbooks - like this one in Stinson - cannot be directly applied.
EDIT:
Here is more formal and simplified version of my question: $A$ is set of all random tapes (or private keys or something), $B$ is is set of all functions from, lets say, $\mathbb{Z}_q$ to $\mathbb{Z}_q$. $\mathcal{A}$ is some algorithm that has parameters $\omega \in A$ and random oracle $\mathcal{O}$ from $B$. We pick elements from $A$ and $B$ with uniform distribution.
For example, assume I want to calculate the probability $ P((\omega, \mathcal{O}) \in A\times B| \mathcal{A}(\omega, \mathcal{O}) \text{ succeeds}) $
This can be rewritten as
$ P((\omega, \mathcal{O}) \in A\times B| \mathcal{A}(\omega, \mathcal{O}(u_1), \ldots, \mathcal{O}(u_Q)) \text{ succeeds}) $
And then I have a Stinson's theorem which can be replaced by some other other "textbook" or citeable theorem.
The main goal is to prove $(\omega, \mathcal{O}(u_1), \ldots, \mathcal{O}(u_Q))$ has uniform distribution by citing something.