I stopped reading near the beginning because either you haven't framed your question as you intended or you have made it overly complicated. The entropy of a discrete distribution depends only on its probabilities--check the formula!--and so your restriction to a bounded set strongly indicates you are considering something other than entropy as commonly understood. Moreover, the maximum entropy solution doesn't exist: it's an "infinite uniform" distribution on the set. The associated expectation (which does depend on the set) would be the limit of the arithmetic mean of its values.

@whuber I added examples to clarify my statements. Is this better?

I can't make sense of this. You lost me where you used the symbol "$X$" to refer both to a probability distribution and random variables, as well as the vague and confusing description that follows about "supports" and "equal-length intervals." You are fighting two fundamental difficulties here: first, you haven't been able to describe in meaningful terms what property "non-uniformity" might be; and second, to cope with that you have resorted to abstract mathematical definitions, but they just don't seem to work. Is there any way you can connect this to standard statistical terms and concepts?

Your definition of 'sample' is unclear *"As s increases by one we can add more than 1 element from A" what does that mean?

For example if the $S_s=\left\{\frac{1}{n}:n\in\mathbb{N},1\le n\le 2s\right\}$. Note that every time $s$ increases by one we add $2$ elements from $A=\left\{\frac{1}{m}:m\in\mathbb{N}\right\}$ to $S_s$ and as $s\to\infty$ we end up covering all elements of $A$

@SextusEmpricus Sorry for the delay, I wanted to make sure everything in my post was clear.

The numerous minor edits, which keep bumping this thread, have become annoying. Expect more downvotes if this keeps up.

I don't get it. $\lim_{n\to\infty}S_n\to\infty$ because that is a harmonic sum, and the harmonic sum is not bounded above.

@Carl Apologies, $S_n$ isn’t a partial sum. Think of it as a set; and we are including new elements instead of adding them.

OK, but, your questions are whether it works and is optimal. Why should it work? I don't see the mechanism whereby it is working; please explain. As for optimal, at the moment it seems arbitrary. Would you please motivate what you are doing enough to characterize it?

@Arbuja is it correct if I change your definition of a sample into the following shorter description: The sample $S(n)$ is a sequence of $n$ subsets $S_i$ of $A$ with increasing cardinality such that $S_i \in S_j$ if $i < j$?...

... what confuses me in your definition is that I am not sure whether the 'sample' refers to the sequence $S(n)$ or to the elements of that sequence $S_i$. This is because your $S_i$ refers to sets, e.g you write in an example $S_2=\left\{1,1/2,1/3,1/4\right\}$ and it refers to the elements instead of to the sequence and the idea that the cardinality needs to increase....

...But with $S_s$ when you replace the particular number in the subscript with a variable $s$ and you seem to indicate that the sample is the entire sequence $S_1, S_2,\dots ,S_s$ generated by the formula expression for $S_s$. I think that it would be useful to be clear with the notation by using $S(n)$ and $S_n$ for the different objects.

the arithmetic mean of f mapped on these samples this is still unclear. What is this?

@SexticusEmpricus Since $S_s$ is finite we can take the arithmetic mean of $f(S_s)$ for every $s\in\mathbb{N}$ as $s\to\infty$. I also made edits to my post.

Please don't attribute to me things I did not write. I noted that your exposition is too abstract and unmotivated to be of interest to anyone on this site (or, IMHO, anywhere else)--and, despite numerous edits and comments, that remains the case.

Re the initial example: in what sense are you conceiving of $\mathbb Q^2$ as a "subset of a finite interval"? You must be using at least one of these words or symbols in an unconventional way.

@whuber I meant $\mathbb{Q}^2\cap[0,1]$.I also edited my post. Hopefully, this time its more clear.

@SexticusEmpiricus I made edits. Hopefully this is better.

The intersection of those two sets is empty. Therein lies the inherent problem of your approach to asking questions here on CV: either you must express them in terms of the substantive statistical problem you face or you must express them completely correctly mathematically. Despite many efforts, you have not done either. Until your strategy changes, I fear you will find your efforts futile and frustrating.