Consider a stochastic process $(X_t)_{t \in T}$ where $T\subseteq[0,\infty)$. The natural filtration $\mathscr F^X=(\mathscr F_t^X)_{t \in T}$ is defined by $$\mathscr F_t^X=\sigma\{X_s^{-1}(B)|0\lt s\le t,B\in\mathscr B(\Bbb R) \}$$
In a book I have read, the author wants to define a mapping $$\psi:\omega\mapsto (X_t)_{t\in T}(\omega):=(\omega(t))_{t\in T}$$ from the underlying probability space $(\Omega,\mathscr A, P)$ to a measure space $(\Bbb R^T, \mathscr S_{cyl})$, the space of sample paths with the sigma-algebra generated by the cylinder-sets. Thus, $$\Bbb R^T=\{f:T\to \Bbb R\}\qquad\mathscr S_{cyl}=\sigma \{f∈\Bbb R^T:\forall i=1,2,...,n,\ f(t_i)\in B_i\}$$ and $$X_t:(\Bbb R^T, \mathscr S_{cyl})\to(\mathbb R,\mathcal B(\mathbb R)),\quad X_t(f)=f(t)$$ is the coordinate mapping in order to use the Kolmogorov Existence Theorem to construct a probability distribution between the time instance $0\le t_1\le\cdots\le t_n\le T$. Equivalently they use a separable process and define the natural filtration $\mathscr G=(\mathscr G_t)$ as the sigma algebras $\mathscr G_t$ generated by the following cylinder sets $$A=\{\omega \in \Omega:\forall i=1,2,...,n,\ X_{s_i}(\omega) \in B_i\}$$
for $0\le s_1\le.....\le s_n \le t$.
My questions are:
1.Where is the natural filtration $\mathscr G$ defined? On $(\Bbb R^T, \mathscr S_{cyl})$? Where is the equivalence to the usual definition $\mathscr F^X$ given above?
2. If I want to calculate the expectation value $E[X^*_t]$ with $$X^*_t=\sup_{0\le t\le T} X_t$$ am I calculating it on $(\Bbb R^T, \mathscr S_{cyl})$ or $(\Omega,\mathscr A, P)$? I am confused between this pair of spaces and how to use them.
Now that did performed a great editing work on your post (I added 2 minor changes to your post on notations).
I am able to give you an answer to the first question, of course $\mathcal{G}$ lives on $\Omega$, does it needs the filtered space $(\Bbb R^T, \mathscr S_{cyl})$ to be defined ? The answer is indeed !
To show you this and answer the last part of your first question let me write a plain generator set $A\in \mathcal{G} $ again departing from your definition and noting that there is an abuse of notation here :
$$A=\{\omega \in \Omega:\forall i=1,2,...,n, s_i \in [0,t]\cap T, \ X_{s_i}(\omega) \in B_i\}$$
Note that $\ X_{s_i}(\omega)$ has no meaning unless you identify (which is done implicitly) $\omega$ with $\psi(\omega)$, so getting back to the definition we get : $$A=\{\omega \in \Omega:\forall i=1,2,...,n, s_i \in [0,t]\cap T,\ X_{s_i}(\psi(\omega))\in B_i\}$$
Ok now we get a clearer picture, to get $\mathscr F_t^X$, you have to use a measure theoretical result but on a collection of set that generates this filtration. But let's suppose that we know that it is generated by sets B (let's call them cylindrical sets ) which can be defined like this :
$$B=\{\omega \in \Omega:\forall i=1,2,...,n, s_i \in [0,t]\cap T,\ \omega \in X^{-1}_{s_i}(B_i)\}$$ here again $X_{s_i}$ is assimilated to the product $X_{s_i}\ (\psi(.)) $ so $X_{s_i}^{-1}(.)= \psi^{-1}(X_{s_i}^{-1}(.))$
So if we want to write things without any implicit notations we get :
$$B=\{\omega \in \Omega:\forall i=1,2,...,n, s_i \in [0,t]\cap T,\ \omega \in \psi^{-1}(X^{-1}_{s_i}(B_i))\}$$
Now compare sets $A$ and $B$ those are exactly the same sets, given the looked for equivalence, as long as you know that the sets of the form $B $ (or $A$) generate the natural filtration (I still need to find the proper reference for this but it is usaly proven with Monotone Class Theorem argument unless mistkan which is a bit heavy for me so I pass).
For the second question, you certainly calculate this over $\Omega$ but to achieve any calculus of this type, you (always) use implicitly the transfer theorem (this a french appellation I don't know the correct name for this in english so please edit this if you know the correct denomination in english) to get for example the image measure form $\Omega$ to $\Bbb R$ measure with respect to Lebesgue measure to express the law of $X^*$.