All the sources I've checked speak of 'a collection', say F , of sets from some set X, and then go on to write things like:If F ∈ F then F c ∈ 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 ∈ instead of ⊂ when referring to members in the collection, OR, is there something more fundamental about collections that I'm missing?

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All the sources I&#039;ve checked speak of &#039;a collection&#039;, say       F  , of sets from some set   X, and then go on to write things like:If   F  ∈      F   then       F    c    ∈  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   ∈ instead of   ⊂ when referring to members in the collection, OR, is there something more fundamental about collections that I&#039;m missing?

Briana Grimes · Accepted Answer

Talking about sets of sets can get confusing, so people instead use the words &quot;collection&quot; or &quot;family&quot; to describe sets of sets (so instead we say &quot;collection of sets&quot; or &quot;family of sets&quot;). This isn&#039;t just in measure theory. I&#039;ve seen this in various math books. The difference is only a matter of connotation, not denotation.As for the usage of   ∈ vs   ⊂, this is not a matter of convention, and I think you&#039;re confused about something there.If A is a set and   B is a subset of   A, we write   B  ⊆  A to denote this. However, if       A   is a set that contains   A as an element, that&#039;s not the same thing as   A is a subset of       A  . No,   A is an element of       A  . You shouldn&#039;t confuse the two notions. You must write   A  ∈      A   because in this scenario   A is itself an object of       A  .

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