Let $X$ be a finite set. Let $M \subseteq 2^X$ satisfy the following properties:
PROPERTY 1:
If $A \in M$ and $A \subseteq B \subseteq X$ then $B \in M$.
PROPERTY 2:
If $A,B,C,D \subseteq X$ satisfy the following conditions:
- $A \cap B = \emptyset$
- $C \cap D = \emptyset$
- $A \cup B \notin M$
- $A \cup C \in M$
- $B \cup D \in M$
Then $C \cup D \in M$.
I wonder if it's always possible to find a function $f: X \rightarrow (0,\infty)$ and a number $\varepsilon >0$ such that:
$$M=\{A \subseteq X: \sum_{x \in A}f(x) \geq \varepsilon\}$$
Showing that these properies are neccessary is fairly easy, but I have no clue how to show that they're sufficient.