# Questions on concepts with multiple definitions

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?

• Personally, I find it handy to keep a list of links to TFAE theorems for big definitions in my field. Then I can point OP in the right direction if it turns out he's using a definition that's different from mine. – Alexander Gruber Nov 27 '17 at 4:13
• This is one of the reasons why it is good to provide context in questions, and definitions being used are often part of the relevant context. – Michael Greinecker Nov 27 '17 at 8:32
• Answerers can not read the minds of askers to discern precisely which of perhaps many definitions the asker has been exposed to. So I agree with @MichaelGreinecker, that it is the asker's responsibility to define the terms they are asking about. Of course, many such questions receive comments to that effect, so the asker often has prompts to define concepts/definitions as they are currently familiar with. – amWhy Nov 27 '17 at 17:49
• I normally ask for context from the OP and sometimes get a reply about the definition being used. When no such context is provided even after request I generally preface the answer with the definition being used. And the option to downvote for lack of context is always available. – Paramanand Singh Nov 29 '17 at 12:25
• Of course you have to be aware that there are different definition in order to take that possibility in account. I for one had not been aware before reading your question that anyone defines $\overline A$ not as the smallest closed set containing $A$ (the intersection is just another formulation of the same definition, of course). – celtschk Nov 30 '17 at 6:59
• @AlexanderGruber: What are TFAE theorems? – celtschk Nov 30 '17 at 7:02
• @celtschk TFAW stands for The following are equivalent" – eepperly16 Nov 30 '17 at 7:51
• It is obvious that answerers cannot read the questioner's mind. In some cases, though, if an answerer happens to be aware that multiple definitions are used in various contexts, they might indicate this awareness rather than blithely answering from a particular perspective. That's only if they happen to be aware of other perspectives, and happen to remember about them at the time, of course. – G Tony Jacobs Dec 2 '17 at 21:09