How to best characterize a monotonic real valued function defined on a power set? The question is pretty vague because it arises from an application scenario and is open-ended.
$\mathcal{S}$ is a countable infinite set, $f$ is a function defined on the power set of $\mathcal{S}$, mapping any subset of $\mathcal{S}$ to a real number in $[0, 1]$. $f$ is known to satisfy the following properties:

*

*For any $S_0 \subseteq S_1$, $f(S_0) \leq f(S_1)$.

*For any $S$ with $|S| \leq 1$ ($|S|$ being the carnality of $S$) we have $f(S) \in \{0, 1\}$.

*There exists infinitely many $S$ (not depending on $f$) such that $|S| = 2$  and $f(S) = 1$.

*There exists a set $S_T$ (not depending on $f$) such that $|S_T| = \infty$ and $f(S_T) = 0$.

What's the best way to "characterize" $f$? i.e., what extra information (as minimal as possible) is needed to determine or construct $f$? For example, it would be great if we can say $f$ is uniquely determined by (or can be constructed from) $f(S)$ for any $S$ with $|S| = 3$ (this is not necessarily true, just an example of a helpful result).
 A: In the comments, you clarified that it would be helpful to have a way to extend a partially-defined function to all of $P(\mathcal{S})$. There is in fact a "universal" way to do this!
Let $\mathcal{P}$ be any subset of $P(\mathcal{S})$ which contains all the subsets of cardinality at most $2$. Then let $f_0 : \mathcal{P} \to [0,1]$ be a function satisfying the following conditions:
(i) For any $S_0 \subseteq S_1 \in \mathcal{P}$, $f_0(S_0) \leq f_0(S_1)$
(ii) For any $S \in \mathcal{P}$ with $\lvert S \rvert \leq 1$, we have $f_0(S) \in \{0,1\}$
(iii) There exist infinitely many $S \in \mathcal{P}$ with $\lvert S \rvert = 2$ and $f_0(S) = 1$
(iv) There exist infinitely many $S \in \mathcal{P}$ such that $S$ has maximal cardinality among elements of $\mathcal{P}$ and $f_0(S) = 0$, OR $\mathcal{P}$ has no element of maximal cardinality but has an ascending chain $S_0 \subset S_1 \subset \dots$ such that $f_0(S_i) = 0$ for all $i$
Then the function $f : P(\mathcal{S}) \to [0,1]$ defined by
$$f(S) = \sup_{\substack{S' \in \mathcal{P} \\ S' \subseteq S}} f_0(S')$$
extends $f_0$ to all of $P(\mathcal{S})$ and satisfies the desired properties.
