I'm trying to express a function greater than zero defined on countable sets dense in $\mathbb{R}$ that are the quotient of countable sets dense nowhere.
Consider $P:A\to\mathbb{R}$ where $P(x)> 0 \text{,} \ $ $\mu(A)=0$ and
$$P(x)= \begin{cases} P_1(x) & x=A_1\\ P_2(x) & x=A_2\\ \end{cases}$$
where $P_1,P_2:\mathbb{R}\to\mathbb{R}$, $A_1,A_2$ are pairwise disjoint and
$$A_1=\left\{\frac{s_1}{t_1}:s_1\in S_1,t_1\in T_1\right\}$$
$$A_2=\left\{\frac{s_2}{t_2}:s_2\in S_2,t_2\in T_2\right\}$$
such that $S_1,S_2,T_1,T_2:\mathbb{Z}^{n}\to\mathbb{R}^{n}$
Does this make sense? If not how to we clarify?
Edit
I'm redoing the definition
Consider $P:A\to\mathbb{R}$ where $P(x)> 0 \text{,} \ $ $\mu(A)=0$ and
$$P(x)= \begin{cases} P_1(x) & x=A_1\\ P_2(x) & x=A_2\\ \end{cases}$$
where $P_1,P_2:\mathbb{R}\to\mathbb{R}$, $A_1,A_2$ are pairwise disjoint and $A_1,A_2$ are countable.
No, it doesn't make sense to me. Unfortunately my suggestion is to rethink what you want to say and rewrite it from scratch.
To clarify, start from the other end. Give the final object you are interested in last, and give auxiliary objects first. Be clear with quantifiers and freedom. Mention the type (function, set, number) of every object you introduce. And most importantly, don't use notation you haven't defined yet.
Can the functions $P_1,P_2,S_1,S_2,T_1,T_2$ be anything? Do they depend on each other? What is the set $A$? How can a number $s_1$ belong to a function $S_1$?
Something like this is much easier to read: "Let $A_1,A_2\subset\mathbb R$ be two disjoint sets. Let $f_i:A_i\to A_i$ be arbitrary functions so that [condition] holds. Then let $A=A_1\cup A_2$ and define $f:A\to A$ so that [condition]."
You can take more freedom in exposition later when you have mastered the basics. But first stick to always defining first and using later.
Comments on the updated version
The new version is better but still unclear. Are $A_1$ and $A_2$ subset of the real line? Is $A$ the union of the two? What is $\mu$? Are the functions $P_1$ and $P_2$ arbitrary or do they have to satisfy something?
The way it appears to me, you have a set $A\subset\mathbb R$ written as the disjoint union of $A_1$ and $A_2$. You have functions $P_i:A_i\to\mathbb R$ which need to be pointwise positive. You have a function $P:A\to\mathbb R$ for which $P_i=P|_{A_i}$. If $\mu$ is the Lebesgue measure, then $\mu(A)=0$ is automatic if both $A_i$s are countable, so I wonder why it's worth a separate mention.
To avoid turning this into a long discussion, I will not comment on any versions beyond the second one. If you want further comments, please ask further separate questions. (You are likely to get more useful feedback if you describe what you want to do in words.)