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My personal notes often have big lists of related, sometimes only suspected, results. What I have done throughout the years in MSE is to ask whether one specific result is true at a time (one post per result). I was wondering if I could write big-list posts containing a large number of some real and some suspected results. I write an example below. $\def\AAA{\mathcal{A}} \def\BBB{\mathcal{B}}$

It'd have the following advantages:

  • It's easier for me :D
  • When answering whether a conjecture holds, it is useful to know, and hence to include in the post, (many) theorems with similar hypotheses.
  • Answering whether a conjecture holds opens a door to solving, and hence it is convenient to inclulde in the same post, similar conjectured results.
  • Other users can access a big-list of related results when needed.

In summary: can I make a post such as the following in MSE?


(Everything that follows is a post example)

I've been studying equivalent characterization (under certain conditions) of measurable functions. Below are a list of (possible) results, each of which is of the form

Result: Given a function $f:(X,\AAA)\to (Y,\BBB)$ between measurable spaces (with some extra conditions), the following are equivalent:

  1. $f$ is measurable.
  2. $f$ complies with condition X.
  3. (suspected!) $f$ complies with condition Y.

The result above is read as follows: conditions $(1)$ and $(2)$ are indeed equivalent, in which case a proof is provided. Condition $(3)$ is merely suspected of being equivalent to the remaining conditions (yet I have not been able to (dis)prove it), in which case I explain why I hold such suspicion.

(Equivalent Characterizations of Borel Measurable Functions) Given a function $f:(X,\BBB_X)\to(Y,\BBB_Y)$ between topological spaces $X$ and $Y$ with $\sigma$-algebras $\BBB_X$ and $\BBB_Y$ generated by the topologies on $X$ and $Y$ respectively, the following are equivalent:

  1. $f$ is measurable.
  2. $f^{-1}(U)\in\BBB_X$ for every open set $U$ in $Y$.
  3. $f^{-1}(K)\in\BBB_X$ for every closed set $K$ in $Y$.

Proof: it is sufficient to note that either the open or closed sets generate the Borel $\sigma$-algebra.

(Equivalent Characterizations of Measurable Functions to a Metric Space) Given a function $f:(X,\AAA)\to(Y,\BBB_Y)$ to a metric space $(Y,\BBB_Y)$ where $\BBB_Y$ is the $\sigma$-algebra generated by the topology of $Y$, the following are equivalent:

  1. $f$ is measurable.
  2. $f$ is the pointwise limit of measurable functions $f_n$.
  3. (suspected!) $f$ is the pointwise limit of measurable simple functions $f_n$.

Proof: that $1\implies 2$ is clear; the other direction is shown here. I suspect $(1\iff 3)$ holds as it is true for functions $f:\mathbb{R}^d\to[0,\infty]$; for a proof see Tao's An Introduction to Measure Theory Lemma 1.3.9 (ii).

(Equivalent Characterizations of Measurable Functions from a Complete Measure Space to a Metric Space) Given a function $f:(X,\AAA)\to(Y,\BBB_Y)$ from a complete measure space $(X,\AAA,\mu)$ to a metric space $(Y,\BBB_Y)$ where $\BBB_Y$ is the $\sigma$-algebra generated by the topology of $Y$, the following are equivalent:

  1. $f$ is measurable.
  2. (suspected!) $f$ is almost everywhere the pointwise limit of measurable functions $f_n$.
  3. (suspected!) $f$ is almost everywhere the pointwise limit of measurable simple functions $f_n$.

Proof: that $(1\implies 2)$ is again clear. My suspicion that $(1\iff 2)$ and an attempt at a proof is given in this post. That $(1\iff 3)$ holds for functions $f:\mathbb{R}^d\to[0,\infty]$ is shown as well in Lemma 1.3.9 (iii) of Tao's book.

...[I could go on, but I believe by now it is clear what I want to do]

  • How can we (dis)prove any of the suspected characterizations?

  • Are there other useful characterizations beyond the ones mentioned?


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    $\begingroup$ As I understand it, the big-list tag is for single questions expecting a big list of answers, not for posts with a big list of questions. Moreover, it's expxlicitly expected that a question post will contain exactly one question, no fewer and no more. So, I think your proposal is a non-starter. $\endgroup$ Commented May 7 at 22:50
  • $\begingroup$ @GerryMyerson I see. What about a post that asks, for example, for equivalent characterizations of measurable functions and as an answer I write the results I know and the ones I suspect? Would that still be innapropriate? $\endgroup$
    – Sam
    Commented May 8 at 8:13
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    $\begingroup$ If it's just a sneaky way to stuff a dozen questions into a single post, then, yes, it would still be inappropriate. If it's an honest attempt to collect equivalent characterizations of measurable functions, that could pass as a big-list question. I think if I did something like that, I'd include the answers I already know in the body of the question, rather than posting them as answers. By the way, I should point out that I'm just some guy, nothing I write here should be taken as an official statement on behalf of math.stack. $\endgroup$ Commented May 8 at 13:33

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No. Each post should contain only a single question.

This site is designed as a place where people can collaborate to build an archive of knowledge, in the form of questions and answers. Much of the site is based on the assumption that there is a single question per post. Various stuff works less well when you have multiple questions in the post. For instance, if someone posts an answer, then the question will be treated as answered, even if only one of the multiple questions have been answered. If one of the questions is clear and another is unclear, then that raises ambiguity about whether to vote to close. etc.

Instead, I suggest that you ask separately about each particular question you have.

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