Condition 1: Let $\mu_{1}$ and $\mu_{2}$ be measures over a non-null support $A$1), $J(\cdot)$ be a function defined over any (degenerate or non-degenerate) measures over $A$, and $\tilde{\mu}$ be a convex combination of $\mu_{1}$ and $\mu_{2}$. If $J(\mu_{1})\leq J(\mu_{2})$ then $J(\tilde{\mu})\leq J(\mu_{2})$.
Condition 2: Let $\mu$ be a measure over a non-null support $A$. $J(\mu)\leq \sup_{a\in A}J(\mu(a))$, where $\mu(a)$ is a degenerate measure that has all the mass at the point $a\in A$ (again, with a slight abuse of the math language).
1) Roughly speaking; can always write this in a more formal way by defining a probability space but that's not essential.
I feel that Condition 1 implies Condition 2, the idea being that we can always start with $\sup_{a\in A}J(\mu(a))$ and iterate using Condition 1 over $A$ to get the desired result, i.e., Condition 2. But I was struggling in writing down a formal proof. And I'm not sure if additional regularity conditions are needed. Any thoughts?