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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?

MJD
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TonyK
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    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. – CyclotomicField Dec 28 '21 at 01:56
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    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. – Arturo Magidin Dec 28 '21 at 02:01
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    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. – Maximal Ideal Dec 28 '21 at 02:08
  • “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. – littleO Dec 28 '21 at 02:33
  • If you use the word ensemble, that is French for set. LOL – shawn_halayka Dec 28 '21 at 02:57
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    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. https://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. – 311411 Dec 28 '21 at 03:54
  • @311411 I consider "equivalence class" to be its own term separate from "class", but I see what you're saying. Perhaps I didn't think things through well enough. – Maximal Ideal Dec 28 '21 at 16:34
  • For the same reason mathematicians sometimes use different types of brackets (e.g. (), [], {} and <>) to delimit different levels of sub-expressions, while in principle only one kind is needed. – Ron Inbar Oct 21 '22 at 15:08

1 Answers1

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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 $\mathcal{A}$.

Maximal Ideal
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  • Confused is my middle name, but that last paragraph helps a lot - I think I've got it now. – TonyK Dec 28 '21 at 02:38
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    @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 ${0} \subseteq {0,1}$. – Jannik Pitt Dec 28 '21 at 10:13