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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:

  1. (boundary conditions) $\mu (\{\emptyset\}) = 0$ and $\mu(X) < +\infty$
  2. (monotonicity) $A_1,A_2 \in \mathcal{F}: A_1 \subset A_2 \Rightarrow \mu (A_1) \leq \mu (A_2)$
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  • $\begingroup$ What do you mean by "continuous," precisely? $\endgroup$ – Math1000 May 17 '18 at 19:24
  • $\begingroup$ Obviously the axiom "being continuous" is sufficient. You might want to be more specific. $\endgroup$ – user251257 May 18 '18 at 19:12
  • $\begingroup$ Do you mean I should provide the convergence and continuity definitions for set-functions? $\endgroup$ – Cebiş Mellim May 19 '18 at 8:49

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