# How should I rewrite the beginning of the linked post?

How should I rewrite the beginning of this post?

It seems people aren’t commenting since it’s difficult to understand.

Any suggestions are welcomed.

Maybe break up the first paragraph, something like this:

Does there exist a function $$f:[0,1]\to[0,1]$$ such that

1. the graph of $$f$$ is dense in $$[0,1]\times[0,1]$$, and

2. the collection of all the pre-images of each of the subsets of the range of $$f$$ under $$f$$, such that the pre-images are non-measurable in the sense of Caratheodory, is a [non-perfect dense set][1] in the collection of all subsets of $$[0,1]$$, and

3. $$f$$ is non-uniform (i.e. without [complete spacial randomness][2]) in $$[0,1]\times[0,1]$$?

I had trouble parsing 2), I don't know whether what I have written conveys what you intended.

• What you’ve written is correct. Commented May 16, 2023 at 12:35
• You stated you have trouble parsing 2. What is your interpretation of this? Commented May 16, 2023 at 12:36
• The way I wrote it reflects my interpretation. I'd say, you look at the range of $f$ – let's call it $R$; you look at a subset of $R$, let's call it $S$; you look at $T$ defined by $T=f^{-1}(S)$; you insist that $T$ be "non-measurable in the sense of Caratheodory" (I don't know what that means, but that's my problem, not yours); and then you ask that the collection of all such $T$ be a non-perfect dense set in the collection of all subsets of $[0,1]$. I guess it's not clear to me what topology you are putting on the collection of all subsets of $[0,1]$. Commented May 16, 2023 at 13:04
• Did you click the link on “non-perfect dense set”. It should lead you to Dave L. Renfro’s answer. Perhaps the topology can be specified there. Commented May 16, 2023 at 13:06
• No, I didn't click on the link. I'm not actually interested in the question, I'm just trying to help you get it to where it might be reopened. Commented May 16, 2023 at 13:11
• As a last note, I added that $f$ should be “measurable in the sense of caratheodory”? Is the question clear or should I make changes. Commented May 16, 2023 at 13:24
• I think it reads much better now than it did before, but I'll leave final judgement to those who have expertise in the subject matter. Commented May 16, 2023 at 13:43