Is collection of subsets of $\Omega$ that are determined by first n number of coin tosses a $\sigma$-algebra?

Question

Let $F_n$ be the collection of subsets of $\Omega$ whose occurrence can be decided by looking at the first n tosses. How can I show $F_n$ is a $\sigma$- algebra?

Answer 1

So, $\Omega$ is the infinite product $\Pi_{i\in \mathbb{N}} \{H, T\}$. The sets you're looking at are such that if $x \in A \in F_n$ then $y \in A$ if $x_i = y_i$ for all $i <= n$.

These are finite collections of sets. You should be able to find most of the proof yourself. I would suggest proving that they're an algebra and then using something that says "finite algebras are $\sigma$-algebras".