I'm trying to get some intuition for the role of the Borel $\sigma$-algebra with an example, but I'm not sure how to interpret parts of this example using math.
Setup
From my notes, a function $X: (\Omega,\mathcal{F}) \rightarrow (\mathbb{R},\mathcal{B})$ is a measurable function if
\begin{equation} X^{-1}(B) = \{ \omega \in \Omega: X(\omega) \in B \} \in \mathcal{F}\ \text{for all}\ B \in \mathcal{B}. \end{equation}
$\mathcal{F}$ is a $\sigma$-algebra using $\Omega$, and $\mathcal{B}$ is the Borel $\sigma$-algebra.
Example
I'm trying to make sense of this definition with a die roll. Let's say you roll a fair die once, so the set of possible outcomes is $\Omega = \{1,2,3,4,5,6\}.$ $\mathcal{F}$ is the set of all subsets of $\Omega$ ($\mathcal{F}$ has other propeties too), so it has sets like $\{\{1\},\{1,2\},\dots,\{1,2,3,4,5,6\},\dots\}.$
Question
What's the "physical" intuition of $\mathcal{F}$ here? If we're only rolling the die once, I don't see why we need to list all these other sets.
As such, what does the Borel $\sigma$-algebra hold, then? From how I understand things now, it would just have numbers like $\{\{1/6\},\{2/6\},\dots,\{1\},\dots\}$, but I think I'm misunderstanding something. Thank you!
The elements of $\mathcal F$ are "events" in the sense of probability theory. So, for example, a gambler might win if he rolls $2$, $4$, or $6$, but lose otherwise. So we need to know about the set $\{2,4,6\}$. In particular, we may want to know its probability $\mathbb P(\{2,4,6\}) = 1/2$. Or we may want to do expectations involving this event, and so on.
The sigma-algebra $\mathcal F$ serves as the domain for the probability measure $\mathbb P$. We are only slightly interested in $\mathcal F$. We are much more interested in $\mathbb P$.