What additional axioms do I have to assume in order to make the set-fucntion in the following definition continuous (and convex)?
Non-additive measure Let $\left(X, \mathcal{F} \right)$ be a measurable space. We then call a set-function $\mu$ a non-additive measure $$\mu_{s}: \mathcal{F} \to [0,+\infty)$$ iff it satisfies following properties:
- (boundary conditions) $\mu (\{\emptyset\}) = 0$ and $\mu(X) < +\infty$
- (monotonicity) $A_1,A_2 \in \mathcal{F}: A_1 \subset A_2 \Rightarrow \mu (A_1) \leq \mu (A_2)$