measurability of a sum of random variables Let $\{X_n\}_{n = 1}$ be a sequence of random variables.
Let $Y_n = \sum_{i=1}^n X_i$
Then, I want to show that $\sigma(Y_1,Y_2,\dots,Y_n) = \sigma(X_1,X_2,\dots,X_n)$ for every $n$.
It is clear to me that $\sigma(X_1,X_2,\dots,X_n) \subset  \sigma(Y_1,Y_2,\dots,Y_n)$. But, how do I show the inclusion in the other direction?

This says


*

*The occurrence or non-occurrence of every event whose occurrence or non-occurrence is determined by the values of $X_1,\ldots,X_n$ is determined by the values of $Y_1,\ldots,Y_n$; and

*The occurrence or non-occurrence of every event whose occurrence or non-occurrence is determined by the values of $Y_1,\ldots,Y_n$ is determined by the values of $X_1,\ldots,X_n$.


Most of the work of proving this is accomplished by showing that (putting it informally) if you know the values of $X_1,\ldots,X_n$ you can find $Y_1,\ldots,Y_n$ and if you know the values of $Y_1,\ldots,Y_n$ you can find $X_1,\ldots,X_n$.
Somewhat more precisely: for every $n$-tuple of values of $X_1,\ldots,X_n$ there is only one corresponding $n$-tuple of values of $Y_1,\ldots,Y_n$, and vice-versa.
