How can I rigorously define the probability space for an infinite sequence of partial sums? Let $x_1, \dots, x_N \in \mathbb{R}$ be $N$ given numbers and let $B \subset \{1, \dots, N\} $ be a random set with a fixed size $S$ such that each element of $B$ is drawn at random and uniformly without replacement. Let $y = \sum_{i\in B}x_i$ be the random partial sum and
$(y_k)$ be a stochastic process where at each step $k$, $y_k$ is the random partial sum where the sum is calculated over random elements in $B_k$.
Questions
1- How can we define the probability space for this process as $(\Omega, \mathcal{F}, \mathbb{P})$?
2- Should I define an adapted stochastic process to a certain filtration like $(\Omega, \mathcal{F}, \{\mathcal{F}_n\}_{n=1}^\infty, \mathbb{P})$?
3- If so, what would be that filtration?
My try
I would start with my sample space first. Let $S$ is fixed. Then there are $r={N \choose S}$ possibilities for selecting the random set at the $k$-th step, i.e., $B_k$. Then $\Omega=\{B^{(1)}, \dots, B^{(r)}\}$ where $B^{(i)}$ $(i=1,\dots,r)$ is one of the possibilities the random set at each step can take.
My confusions
1- At first step, How can I define $\mathcal{F}_1$ as the $\sigma_1$-algebra?
2- When I move on to the next step my possibilities become $\Omega \times \Omega$, and how I should define $\mathcal{F}_2$ as the $\sigma_2$-algebra?
3- As I keep going and get to let's say the $n$-th step, my possibilities become $\Omega \times \dots \times \Omega$? Then what would be $\mathcal{F}_n$ as the $\sigma_n$-algebra?
4- What happens when $n$ goes to infinity? Can I use the countability of the process and say I can find a $\mathcal{F}=\{\cup_{i=1}^{\infty}\mathcal{F}_i\}$
?
5- When I write $y_k(\omega)$, is $\omega \in \Omega$ a sequence of realizations of random sets? If so, my choice of $\Omega=\{B^{(1)}, \dots, B^{(r)}\}$ is not right because it has to be an infinite set including every order of random sets that are chosen.
Note
As I am sure this would be a general question that every one may face, please answer my questions very clearly and address them one at the time.
 A: Let $E = {\{1,\ldots,N\} \choose S}$ (the set of all subsets of $\{1,\ldots,N\}$ having size $S$), $\mathcal{P}(E)$ its power set and $\mathcal{U}(E)$ the uniform distribution on $E$. The probability space $(E,\mathcal{P}(E),\mathcal{U}(E))$ is sufficient to modelize one choice
at random in $E$.
If you do want to modelize $S$ successive samplings without replacement in $\{1,\ldots,N\}$, you may choose alternatively $E$ equal to the subset of all $S$-uples in $\{1,\ldots,N\}^S$ whose coordinates are all different.
Yet, the additional information (in which order the $S$ integers obtained are sampled?) plays no role in the quantity you study, namely, the sum of $x_i$ over all $i \in B$, where $B$ is the set of all integers sampled.
To modelize infinitely many independent such experiments, a natural probability space is $\Omega = E^\infty = \prod_{k \ge 1} E$, endowed with the product $\sigma$-field $\mathcal{P}(E)^{\otimes \infty}$ and the product probability measure $\mathcal{U}(E)^{\otimes \infty}$.
For $\omega = (\omega_1,\omega_2,\ldots) \in \Omega$, each $\omega_k$ belongs to $E$, so you can set $B_k(\omega) = \omega_k$ and $$Y_k(\omega) := \sum_{i \in \omega_k} x_i.$$
The first `natural' filtration $(\mathcal{F}_k)_{k \ge 1}$ on $\Omega$ is the filtration generated of the coordinate process $(B_k)_{k \ge 1}$. A subset $A$ of $\Omega$ belongs to $\mathcal{F}_k$ if and only if $1_A(\omega)$ is a function of $(\omega_1,\ldots,\omega_k)$ only.
I hope that it helps.
A: From what I understand, all the $B_k$s are independent, and so are the $y_k$s.
A stochastic process is  a collection of random variables defined on a common probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$.
$y_k$ is the $k^{th}$ random variable of this stochastic process, and all $y_k$ are iid, so let's consider one $y_k$ at a time and define the P-space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ for it.
Let $S$ be fixed. Then there are $r={N \choose S}$ possibilities for selecting the random set at the $k$-th step, i.e., $B_k$. Then $\Omega'=\{B^{(1)}, \dots, B^{(r)}\}$ where $B^{(i)}$ $(i=1,\dots,r)$ is one of the possibilities the random set at each step can take.
Then  $\Omega = \left\{ \sum_{j=0}^SB^{(i)}_j  \right\}_{i=1}^{r} \hspace{2em} $ [You could also use $\Omega'$ rather than $\Omega$]
This is a discrete-valued finite set and thus, a possible $\mathcal{F}$ is simply $ 2^{\Omega}$.
You can then easily define a probability measure - here it'll be uniform over $\Omega'$.
I'm not quite familiar with the need for filtrations in this context. This and this might be of help in that case.
