Consider the following excerpt from the book Mathematical Theory of Bayesian Statistics by Watanabe. Right on the second page, the following is stated:
What does the second sentence really mean?
It seems to me that the author wants to transmit an informal idea (but I don't know which one), since if we consider the statement literally, it doesn't make sense: One cannot condition on an a distribution, only on an event.
Why, then, is this "representation" used (and moreover, a misleading term "conditional probability distribution used") if the first sentence already describes in a perfectly clear manner what is going on?
The second sentence means that given the distribution is q, here is the probability of getting the set of data points. It is a very careful way to remind the reader that there is typically an assumed distribution underlying the process in question.
The "event" here is that the process $q$ is present, and not say some process $r$, $s$, or $t$.
At a more general level, one can always add an explicit condition, but we generally do not do so just to clarify notation and simplify a discussion. We could include the condition that the measuring system is the same (or not!).
Explanation: Every statistical measurement, such as concerned here, is based on some assumptions, or elements that can be expressed as "given that...". For instance, when we say that the probability of a coin coming up heads is $0.5$, we often write it as $P(H) = 0.5$. But what we really mean is that the probability is $0.5$ GIVEN THAT the coin is fair. So we might write $P(H|{\rm fair}) = 0.5$. Is that enough? No! We assume that the flipping procedure is fair too... that the flick of the thumb does not favor heads or tails. Thus we could write $P(H|{\rm fair} \& {\rm fair~flip}) = 0.5$. Is that enough? No! We assume that the person reporting the flip is honest, and has good eyesight. Thus we could write $P(H|{\rm fair} \& {\rm fair~flip} \& {\rm accurate~reporting}) = 0.5$. Are we done yet? No!... Well you get the idea.
In statistics we take certain assumptions as given, needing no explicit expression in our equations.
That's all this overall question refers to: Watanabe is making it explicit that the model is $q$. He doesn't have to do this, but it cannot hurt. It is like writing $P(H|{\rm fair}) = 0.5$, above. The "fair coin" would be one possible "model," $q$.
Frankly, OP, I think you're spending far too much time on this issue, and your discussion of "product measures," $\sigma$ algebras, and such is an irrelevant diversion.