Why a 'collection' of sets and not a 'set' of sets in sigma-algebra

All the sources I've checked speak of 'a collection', say $\mathcal{F}$, of sets from some set $X$, and then go on to write things like:

If $F\in\mathcal{F}$ then $F^c\in F$,

and so on.

Is it just convention in measure theory to speak of collections of sets instead of sets of sets and to use $\in$ instead of $\subset$ when referring to members in the collection, OR, is there something more fundamental about collections that I'm missing?

Solution 1:

Talking about sets of sets can get confusing, so people instead use the words "collection" or "family" to describe sets of sets (so instead we say "collection of sets" or "family of sets"). This isn't just in measure theory. I've seen this in various math books. The difference is only a matter of connotation, not denotation.

As for the usage of $\in$ vs $\subset$, this is not a matter of convention, and I think you're confused about something there.

If $A$ is a set and $B$ is a subset of $A$, we write $B\subseteq A$ to denote this. However, if $\mathcal{A}$ is a set that contains $A$ as an element, that's not the same thing as $A$ is a subset of $\mathcal{A}$. No, $A$ is an element of $\mathcal{A}$. You shouldn't confuse the two notions. You must write $A\in\mathcal{A}$ because in this scenario $A$ is itself an object of $\mathcal{A}$.