I left a comment. Here or at mse, it is not clear how he is using the word average, as it comes up in several ways that do not seem compatible.

I was the first to vote to close and I admit I didn't read the question the way Douglas did. I hadn't interpreted it as really looking for a proof in complete generality but more like looking for confirmation that this is not a general phenomenon. I think the tone, the username, etc made me quickly read it as a mathstackexchange type question. I'll vote to reopen.

While I do not vote to close anymore myself, I would still like to go on record as saying that this seems like a prime example of a question that should be sent to math.SE.

The usual conditions that I remember seeing which ensure that an additive function on the reals is linear are continuity or local boundedness. I was pleasantly surprised to see measurability as a condition when I was writing my answer. The question asks what the minimal condition is. Perhaps someone can say whether measurability is essentially minimal.

Closed questions with substantive answers are generally not deleted.

@Ryan: I agree. Some questions should be closed but not deleted. But there are some trigger-happy deleters who need to be convinced of this.

If somebody gives a good answer to a question and starts a thread that the answer should stay open, then I am willing to give them the benefit of the doubt. I also agree that closed should not be equal to deleted. My fear is that some time down the future this particular question may be deleted because it looks like at first sight like one of those obvious delete questions (including being closed as off-topic), in part because the OP is not writing in the style of a professional mathematician. But Douglas found an interesting mathematical question here and if people do not read what he wrote carefully then I can imagine this question being deleted in a month or so after this discussion is forgotten.

For example I remember a question that had 2 delete votes (I never checked whether it was deleted) asking for examples where Cech cohomology differed from derived functor cohomology. It was closed as a duplicate because it was previously asked and answered with an example from schemes. But before the second version was closed, David Roberts gave an example not coming from algebraic geometry which received several upvotes and which answered in some sense better the original version where the OP had asked in particular for examples coming from algebraic topology and not algebraic geometry. So I think that sometime squestions get delete votes with reasonable answers. It obviously is time consuming to really check carefully each closed question to decide if it should be deleted or not.

As Bill said, questions with substantive answers are not to be deleted unless there are egregious circumstances. If you are aware of a question with substantive answer that has been deleted, let the moderators know.

No if the delay is that large my argument does not apply (I'd say 35 days is about the max for it; 2+a little for the first, then 30days on the delete-votes lists). Perhaps 'almost always' was an exageration; I reduce it to 'mainly'. Yet, I agree, there is certainly a risk of oversights for deletetion. For me, because as opposed to closure (for the most part), I do it (when I do it, not so much lately) mainly following a list of question were this (ie, possibly delete them) is the point of looking at them, and not by navigating to the question for another motivation.