I've started learning about measure theory and I'm trying to get some intuitive grasp of the basic concepts. This is only succeeding partially so far.
There is an exercise which I don't quite understand. Here it is:
Let $\Omega$ be the set of all sequences $\omega = (\omega_1,\omega_2,\ldots)$ where $\omega_n \in \{0,1\}$ $\forall n \geq 1$. Define for all $n$ the projections $p_n:\Omega \rightarrow \{0,1\}$ and let $\mathcal{F}_n = \sigma(p_1,\ldots,p_n)$. Prove that $\mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \ldots$
The course material that I'm using defines the $\sigma$-algebra of a function in the context of borel sets $B \in \mathcal{B}(\mathbb{R})$, as in $\sigma(f) = \{\{f \in B\} : B \in \mathcal{B}(\mathbb{R})\}$ where $\{f \in B \} = \{\omega \in \Omega : f(\omega) \in B\}$
I have two questions about this exercise:
1) Is a $\sigma$-algebra generated by some function $f$ always defined in the context of Borel-algebras? Ie, in the case of our functions $p_i$, should we think of $p_i$ as a function mapping some sequence $\omega$ to $\{0,1\} \subseteq (a,b)$ for some $a,b \in \mathbb{R}$?
2) How should I read $\sigma(p_1)$? Because I'm having trouble connecting the nature of $p_i$ with the aformentioned definition of $\sigma(f)$.