Let $X$ be a subset of $\mathbb{R}^n_{++}$ (vectors with non-negative coordiantes), and let $M$ be the set of regular Borel measures with finite variation on $X$ and finite first moment: $$M:=\{\mu;\ |\mu|<\infty,\int_X|x|\text{d}\mu<\infty\}$$
What can be said about the measures in $M$ that satisfy $\int_Xf\text{d}\mu\geq 0$ for all $f:X\rightarrow \mathbb{R}$ such that $f(0)=0$ and are:
- non-negative.
- continuous.
- convex.
- with bounded partial derivative that exists almost everywhere.
(Notice that such a function is always $\mu$-integrable since it is dominated by a linear function outside a compact subset),
Clearly, every non-negative measure belongs to $M$, but what other measures may be in there?