Alternative definitions of stochastic processes?

A stochastic process is defined as a family of random variables $\{X_t: \Omega \to S, t \in T\}$ , where $\Omega$ is a probability space, $T$ is a set, and $S$ is a measurable space. Equivalently, a stochastic process can be defined as a mapping $X: \Omega \to S^T$ such that it is measurable with respect to the sigma algebra on the probability space $\Omega$ and the product sigma algebra on $S^T$.

Now let me try to define a different kind of "stochastic processes", by replacing the sigma algebra on $S^T$ with a different one. Suppose $A \subset S^T$ which is not necessarily measurable wrt the product sigma algebra on $S^T$. Now introduce the smallest sigma algebra $\mathcal M$ on $A$ such that $\forall t \in T$, the mapping $t: A\to S$ defined as $t(a): =a(t), \forall a \in A$ is measurable. Now define a "stochastic process" to be a measurable mapping $Y:\Omega \to A$ wrt the sigma algebras on $\Omega$ and on $A$. An example of such a "stochastic process" is a random measure, where $T$ is a sigma algebra, $S$ is $[0, \infty]$ and $A$ is a set of measures defined on $T$.

My questions:

1. When viewing a "stochastic process" $Y:\Omega \to A$ as a mapping $Y:\Omega \to S^T$, is $Y$ not necessarily a stochastic process? If possible, what conditions can make $Y$ a stochastic process?

For example, can a random measure $Y:\Omega \to A$ be viewed as a family of random variables $\{Y_C: \Omega \to [0, \infty], \forall C \in \mathcal M\}$?

2. Is a stochastic process $X: \Omega \to S^T$ necessarily a "stochastic process"? In other words, on $S^T$, is the product sigma algebra the same as

the smallest sigma algebra on $S^T$ such that $\forall t \in T$, the mapping $t: S^T\to S$ defined as $t(f): =f(t), \forall f \in S^T$ is measurable?

Added: Yes, I think it is. Because product sigma algebra is defined to make projections measurable.

Thanks and regards!