On which measure space is $S_n = X_1 + \dots + X_n$ considered? A common setting in law of large number theories is letting $X_1, X_2, \dots$ be independent indentical random variables on probability space $(\Omega, \mathcal{B}, P)$. Let $S_n = X_1 + \dots + X_n$. My question is are we considering $S_n$ as a random variable


*

* on the product space of $n$ copies of $(\Omega, \mathcal{B})$, so that $S_n(\omega) = X_1(\omega_1) + \dots + X_n(\omega_n)$, where $\omega = (\omega_1, \dots, \omega_n)$, and $\omega_i \in \Omega$ for all $1 \le i \le n$, or


* on $(\Omega, \mathcal{B}, P)$, so that so that $S_n(\omega) = X_1(\omega) + \dots + X_n(\omega)$?


Another related question is that what role does the product measure space of a countably infinite copies of $(\Omega, \mathcal{B})$ play in the theory?
Thanks so much for answering my first question on mathexchange!
 A: In this sort of situation, $(X_n)_{n=1}^{\infty}$ are considered to be simultaneously defined on $(\Omega,\mathcal{B},P)$. So, if we define $S_n:=X_1+\cdots+X_n$, then your second option is the correct one: we think of $S_n:\Omega\rightarrow\mathbb{R}$ by
$$
S_n(\omega):=X_1(\omega)+\cdots+X_n(\omega).
$$
As to your second question... one place that the product measure space comes in handy is this: given a random variable $X$ on some probability space $(\Omega,\mathcal{F},P)$, you can use the countable product of separate copies of $\Omega$, with the product $\sigma$-algebra and product measure, to construct a new space $(\Omega',\mathcal{F}',P')$ and random variables $(X_n)_{n=1}^{\infty}$, $X_n:\Omega'\rightarrow\mathbb{R}$, such that the $(X_n)$ are independent and identically distributed, and have the same (marginal) distributions as $X$.
A: The $S_n$ are defined on $(\Omega,\mathcal B,P)$.
Infinite product measure spaces come into play when you want to show that, given a random variable $X$ defined on, say, $(\Omega_0,\mathcal B_0,P_0)$, it is indeed possible to define a sequence of independent random variables with the same distribution as $X$.
Specifically, let $P_X$ be the distribution of $X$, which is a probability measure on $\mathbb R$. If you define $(\Omega,P)$ to be the infinite product of countably many copies of $(\mathbb R,P_X)$, that is, 
$\Omega=\mathbb R^{\mathbb N}$ and $P=\otimes_0^\infty P_n$ where $P_n=P_X$ for all $n$, then the coordinate functions $X_n(\omega_1,\omega_2,\dots ,)=\omega_n$ do the job.
A: One can regard $X_1$ as having a particular measure space as its domain and $(X_1,\ldots,X_n)$ as having as its domain the $n$-fold Cartesian product of that space with itself, and if one calls that last space $(\Omega,\mathcal{F},P)$, then all of the components are defined on that one space.  But it's not necessary for all that to be true in order to draw conclusions about the distribution of $(X_1,\ldots,X_n)$ when the fact that they are i.i.d. is known.  The fact that they are i.i.d. and the distribution of $X_1$ determine the distribution of $(X_1,\ldots,X_n)$.  For many purposes one can just regard the range of $X_1$ as the probability space on which $X_1$ is defined and the $n$-fold product of that range with itself as the space on which $(X_1,\ldots,X_n)$ is defined.
A: Suggestion. In law of large number theories you always can assume that you work in the following model:
Let $F$ be a distribution function defined in the real axis $R$.  Let $\mu$ be a Borel  probability measure in $R$ defined by $F$. We set $R_i=R$, $B_i=B(R)$ and $\mu_i=\mu$ for $i \in N$, where $N$ is a set of all natural numbers.  We put $(\Omega, F, P):=\prod_{i \in N}(R_i, B(R_i), \mu_i)$ and $X_n((\omega_k)_{k \in N})=\omega_k$ for $n \in N$ and $(\omega_k)_{k \in N} \in R^{\infty}$.   
Then  $( X_k)_{k \in N}$ stands an example of independent identical random variables (with a distribution function $F$)  on probability space $(Ω,B,P)$ and $S_n((\omega_k)_{k \in N})=\sum_{k=1}^n\omega_k$.
