It is known that if $X_i$ are iid, $E(|X_i|) < \infty$ and $E(X_i) = 0$ then $S_n = \sum_1^n X_i$ is a martingale. Suppose all $X_i$ are defined wrt sample space $\Omega$.
I don't understand why $S_n$ is $\sigma(X_1, .. X_n)$-measurable? Specifically, I am confused about the sample space for $S_n$. To be $\sigma(X_1, .. X_n)$-measurable the sample space of $S_n$ should be $\Omega$. I do not see why. Any help would be appreciated.