Recently, I've seen several questions on measure theory in which an answer must prove a set is measurable. To several of these questions, for example this, someone replies in the comments or an answer something to the effect of "$\sigma$-algebras are closed under [some set theoretic operation] so this set is measurable." These answers often strike me as not very useful to the OP as many measure theory classes begin with a construction of the Lebesgue measure and then later discuss abstract measure theory. Thus, it is unlikely that they have already proven the set of Lebesgue measurable sets form a $\sigma$-algebra, so these answers are not helpful to them.
The problem is further complicated by the many logically equivalent definitions of measurability (e.g. Caratheodory, approximation by open sets, etc.), which can also lead to answers that are unhelpful to the OP as they use a different definition of measurability that they may not yet have proved is equivalent to the one they are familiar with.
Similar problems occur in other subjects. For example, I have seen users trying to prove $\bar{A}$ is closed. Using one definition of the closure (the intersection of all closed sets containing $A$), this statement is trivial; using the other common definition ($\bar{A} = A \cup A'$) this question is less trivial.
As I see it, the problem is two-fold:
- Questioners not specifying which of the many equivalent definitions of concept they are using, thus making it hard for answerers to provide helpful answers.
- Answerers assuming the OP is using a logically equivalent definition of a concept which they would be unfamiliar with and then providing answers to the question that aren't helpful to the OP. (Of course, these answers may still be useful to the OP in the future and to other users.)
I'm not really sure there is a fix to this problem, but I thought I would bring it up in meta. Have others seen situations like this? Other than requesting feedback in the comments, is there a way we can make it clear to new users that there often many equivalent definitions of concepts they are using and suggest they include which one they are using in their questions?