# May I ask big-list questions like so?

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'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?

• 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. Commented May 7 at 22:50
• @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?
– Sam
Commented May 8 at 8:13
• 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. Commented May 8 at 13:33