Sigma algebra generated by random variables

Question

Im currently studying for an exam in probability theory and one example in the lecture notes stated the following: Let $X_1,X_2,...$ be i.i.d. random variables. Then $\sigma(\{X_i:i\leq n\}) = \sigma(\{S_i:i\leq n\})$, where $S_n:= \sum_{i=1}^{n} X_i$. I don't see why this statement is true, maybe its some measure theoretic standard argument, but I just don't see it. Any help is really appreciated, thank you in advance!

Answer 1

Write $S_i=f_i(X_1,\ldots,X_n)$ where $f_i:(x_1\ldots,x_n)\mapsto \sum_{k=1}^i x_k$ and note that $S_i^{-1}(A) = (X_1,\ldots,X_n)^{-1}(f_i^{-1}(A))$, so that $\sigma(\{S_i:i\leq n\}) \subset \sigma(\{X_i:i\leq n\})$.

For the other inclusion, note that $X_1=S_1$ and for $i\geq 2$, $X_i=g_i(S_1,\ldots,S_n)$ where $g_i:(s_1,\ldots,s_n)\mapsto s_i-s_{i-1}$.