The $I$ in the smallest sigma algebra generated by collection of subsets of $\Omega$ In defining a Borel sigma algebra (and if I understand it right) you can depart from the idea that an arbitrary collection of subsets $\mathcal C$ of the sample space $\Omega$, where $\mathcal C$ will end up being intervals of the real line, generates  the smallest sigma algebra $\sigma(\mathcal C)$ containing all elements of $\mathcal C$, which happens to also be unique (?).
To find this smallest sigma algebra, I have come across the following notation:
Let $\{\mathcal F_\color{red}{i},i\in \color{red}{I}\}$ be all the collection of all the sigma algebras containing $\mathcal C$. Then $\sigma(\mathcal C)=\bigcap\limits_{i\in I}\mathcal F_i$.
So I think I get the idea, but I don't know what $I$ represents.
SOURCE: here
 A: This is a common notation: Suppose you have a set $\mathscr{C}$, all whose elements are also sets. The you might want to consider the intersection of all elements of $\mathscr{C}$. We might denote this as $\bigcap\mathscr{C}$.
Example 1: $\mathscr{C}=\left\{\left\{1,2,3\right\},\left\{1,4\right\}\right\}$. Then $\bigcap\mathscr{C}=\left\{1,2,3\right\}\cap\left\{1,4\right\}=\left\{1\right\}$.
Example 2: $\mathscr{C}=\left\{\left\{k\in\mathbb{N}:k\leq n\text{ or }k\text{ is not divisible by }n\right\}:n\in\mathbb{N}, n\geq 2\right\}$. Then $\bigcap\mathscr{C}=\bigcap_{n=2}^\infty\left\{k\in\mathbb{N}:k\leq n\text{ or }k\text{ is not divisible by }n\right\}$ is the collection of prime numbers
However, the notation $\bigcap\mathscr{C}$ is not universal, and when dealing with intersections of large collections of sets, some people prefer to use a notation such as $\bigcap_{n=1}^\infty$ or $\bigcap_{n\in\mathbb{N}}$.
So, in general, when we write $\mathscr{C}=\left\{C_i:i\in I\right\}$, we are simply saying that $I$ is some set and the map $I\to\mathscr{C}$, $i\mapsto C_i$ is a bijection. We can always do that in a trivial way: take $I=\mathscr{C}$ and $C_i=i$.
Then we use the notation $\bigcap_{i\in I} C_i$ instead of $\bigcap\mathscr{C}$.

In your particular problem, we consider the class $\mathscr{F}$ of all $\sigma$-algebras containing $\mathcal{C}$ (do not confuse this $\mathcal{C}$ with the $\mathscr{C}$ above). We then define $\sigma(\mathcal{C})=\bigcap\mathscr{F}$. Just as above we are simply writing $\mathscr{F}=\left\{\mathcal{F}_i:i\in I\right\}$, and using the notation $\sigma(\mathcal{C})=\bigcap_{i\in I}\mathcal{F}_i$ instead of $\bigcap\mathscr{F}$.
I do not know if we can calculate the cardinality of $\mathscr{F}$ (which is the same as the cardinality of $I$) in general, or at least it is not obvious how to do so, so we cannot really replace $I$ by any known set.
