Question 70
Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the set $\mathbb{N}$ of positive integers. Find all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties:
(i) if $S \subseteq T$ are non-empty subsets of $\mathbb{N}$, then $f(T) \le f(S)$;
(ii) for all $S, T \in \mathcal{X}$,
$$
f(S) + f(T) \le f(S + T), \quad f(S)f(T) = f(S \cdot T),
$$
where $S + T = \{s + t \mid s \in S, t \in T\}, S \cdot T = \{s \cdot t \mid s \in S, t \in T\}$.
(i) if $S \subseteq T$ are non-empty subsets of $\mathbb{N}$, then $f(T) \le f(S)$;
(ii) for all $S, T \in \mathcal{X}$,
$$
f(S) + f(T) \le f(S + T), \quad f(S)f(T) = f(S \cdot T),
$$
where $S + T = \{s + t \mid s \in S, t \in T\}, S \cdot T = \{s \cdot t \mid s \in S, t \in T\}$.