Suppose we have a sequence of $\sigma$-field generated by random variables $\mathscr F_{n}$, where $\mathscr F_{n}=\sigma(X_{1},\dots, X_{n})$
$(1)$Can we say that $\mathscr F_{n}$ is an increasing sequence of $\sigma$-field, which means $\mathscr F_{n} \subset \mathscr F_{n+1}$?
$(2)$ $\cup_{n}\mathscr F_{n}$ is a field, or a $\sigma$-field? How to prove it?
The answer to the first question is affirmative but not the second.
(1) By definition, $\mathcal{F}_n := \sigma(X_1,\dots,X_n)$ is the smallest $\sigma$-algebra such that $X_1,\dots,X_n$ are all measurable with respect to $\mathcal{F}_n$. If $X_1,\dots,X_n,X_{n+1}$ are measurable with respect to $\mathcal{F}_{n+1}$, then surely so do $X_1,\dots,X_n$, so $\mathcal{F}_{n+1}$ is a $\sigma$-field which $X_1,\dots,X_n$ are measurable with respect to. Since $\mathcal{F}_n$ is the smallest $\sigma$-field with such a property, we must have $\mathcal{F}_n \subseteq \mathcal{F}_{n+1}$.
(2) The axiom that is stopping the union from being a $\sigma$-algebra is the countable union of events being an event as well. Refer to here for a counterexample.
