I am an engineer by training and trying to self-study Martingales by myself. I came across the concept of a natural filtration and I have a conceptual question.
Suppose that $X_1,X_2,X_3,\cdots$ is a sequence of i.i.d. random variables. The natural filtration is understood to be $\mathcal{F}_i = \sigma(X_1,X_2,\cdots,X_i) $.
This definition confuses me. If my understanding is correct, "$\sigma(X_1,X_2,\cdots,X_i)$" is the "smallest sigma-algebra such that each of the random variables $X_j, j=1,2,3,\cdots,i$ is measurable with respect to it."
Since each $X_k$ has the same distribution, dosen't this imply that $\sigma(X_1) = \sigma(X_1,X_2) = \sigma(X_1,X_2,X_3) \cdots $?
While we are on this topic, I read that a random variable is simply a mapping between sample spaces and the real line. So, if all random variables in the sequence $X_1,X_2,X_3,\cdots$ are defined on the same probability triple $(\Omega, \mathcal{F},\mathcal{P})$ and have the same distribution, does this also imply the "functions" or "maps" $X_i :\Omega \to \mathcal{R}$ and $X_j : \Omega \to \mathcal{R}$ are one and the same? Then, what is the meaning of independence?