I am getting confused when it comes to summing iid random variables. Suppose we have $X_1$ and $X_2$ iid random variables where $X_i : \Omega = ${$heads, tails$}$ \to ${$-1,1$} and $X_i(heads) = 1$, $X_i(tails) = -1$, and each have equal probability. Now if we consider $Y = X_1 + X_2$, do we have that $Y$ is defined on $\Omega \times \Omega$, because if it was just defined on $\Omega $ then surely we would obtain theses incorrect results: if we have that $\omega = heads$ then, $$Y(\omega) = X_1(\omega) + X_2(\omega) = 2$$, and if $\omega= tails$ then, $$ Y(\omega) = X_1(\omega) + X_2(\omega) = -2$$
Now if we say that Y has the domain $\Omega \times \Omega$, then how can a simple symmetric random walk be a martingale since each term in the sequence will be defined on a different sample space, and for a martingale, we require the sequence of random variables are defined on the same sample space.