TITLE: Example of filtration in probability theory
QUESTION [21 upvotes]: I'm studying Martingales and before them filtrations. Given a probability space $(\Omega, F, P)$ I define a filter $(F_n)$ as a increasing sequence of $\sigma$-algebras of $F$, such that $F_t \subset F$ and
$t_1 \leq t_2 \Longrightarrow F_{t_1} \subset F_{t_2}$. Here comes my question:
How can the $F_t$'s be $\sigma$-algebras and subsets of $F$ without being exactly equal to $F$? I suppose that $F_t$'s being $\sigma$-algebras mean that they are $\sigma$-algebras with respect to the measure space $(\Omega, F)$.
Can anyone explain why they are not necessarily equal to $F$ and give an example where this is obviously false?

REPLY [2 votes]: This is a variation on @GEdgard example, that can be applied to real valued random variables. The probability plays no role. Let $X_j$ be the projection of $\mathbb{R}^n$ onto the the $j^{th}$ component, i.e.:
$ X_j(x_1,x_2,\dots,x_n) = x_j $. Let $\mathcal{F}_j = $ sigma-algebra generated by $X_1, \dots, X_j$. $\mathcal{F}_j $ is the sigma-algebra of Borel sets that can be defined in terms of the first $j^{th}$ coordinates, in the sense that.:
$$
\mathcal{F}_j = \{A \times\mathbb{R}^{n-j} | A \text{ is a Borel set of }\mathbb{R}^{j}\} 
$$