# 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?

• It's more of a convention to help keep the different levels of nesting clearly delineated. You can get lost in sets of sets of sets of sets pretty easily. Commented Dec 28, 2021 at 1:56
• Same reason we talk about "a family of sets", rather than just saying "a set of sets". It is easier to keep the hierarchies in mind if you have elements<sets<collections, than if you have reals<sets-of-reals<sets-of-sets. Commented Dec 28, 2021 at 2:01
• One thing I didn't address in my answer: there is one word that does not mean the same thing as a set. This word is "class." A proper class (class that is not a set) is like a set, but you can't apply the usual manipulations on them. They were devised to avoid Russel's paradox while still allowing us to talk about "the class of all sets." For all intents and purposes, you won't run into this word unless you study set theory, and outside of set theory you should avoid using that word. Commented Dec 28, 2021 at 2:08
• “Collection” is another word for set. They could say “set of sets”, and in fact I would prefer that, but I guess they think that sounds awkward. Commented Dec 28, 2021 at 2:33
• I am not sure I can join in the banishment of "class" from the family {set, collection, family, ensemble}. Its use is easy to find on this site (e.g. math.stackexchange.com/a/125495 ) and besides do you abandon the phrase "equivalence class"? Russell's paradox is not something I think of very frequently. Commented Dec 28, 2021 at 3:54

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 $$\mathcal{A}$$.
• @TonyK To give an example which makes the second point more explicit: If $\mathcal{A} = \{ \emptyset, \{0\}, \{1\}, \{0,1\} \}$ ($=2^{\{0,1\}}$), then $\{0\} \in \mathcal{A}$, but $\{0\} \nsubseteq \mathcal{A}$. However $\{ \{0\}\} \subseteq \mathcal{A}$ and $\{0\} \subseteq \{0,1\}$. Commented Dec 28, 2021 at 10:13