I am thinking about the following function:
$$ p : A \to \Delta \big( F(x, y(t) ) \times T \big) ,$$
where $t \in T$ denotes continuous time, and $\Delta (X)$ denotes the set of all probability measures over an arbitrary space $X$.
The function $p$ can be seen as one that gives a probability over (1) a space of uncertainty that does change – indirectly through $y(t)$ with $t \mapsto y(t)$ – according to time $t \in T$ and an additional independent variable $x \in X$, and (2) time $T$ itself.
Beyond the fact that the function (and the situation) can be considered a bit unrealistic, is it correct this way of writing if I want to stress that $y(t)$ has to go in the probability measure along with the $t \in T$ that are considered by that probability measure?
Thank you for your time and for any feedback.
As it can be seen by the title, I am completely at loss concerning it (sorry). Any feedback from the moderators concerning it is (really!) more than welcome!
PS: I was not sure about the tag mathematical modelling, but it seems it is the closest one (along with notation) to the kind of problem I am describing.
I'm not sure if this is what you want but perhaps you want to the codomain to be:
The first is more general since it allows correlation across time. The second is more restrictive because assumes independence (and in this case we may have to assume $T$ discrete as there are problems in defining a continuum of independent random variables).