Hi everyone,

my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}_{\infty} ( \Gamma )$, and a convex function $F : A \to [0, + \infty]$ which is lower semicontinuous with respect to pointwise convergence, and it satisfies:

- $F(0)=0$;
- $F( \lambda f ) = \lambda F( f)$ for every $f \geq 0$, $\lambda \geq 0$.
- $F( f) \geq \inf_{\Gamma} f$

Can I find a nonnegative (sigma-additive) measure $\mathfrak{m}$ on $\Gamma$ such that $\int_{\Gamma} f d \mathfrak{m} \leq F(f)$ and $ \mathfrak{m} (\Gamma) >0 $?

[This reminds me a bit Hahn-Banach theorem: recalling Daniell's integral I'm asking a linear functional $L$ that stays between $F$ and $\inf_{\Gamma} f$ (that is concave) and with additional condition that $L(f_n) \to 0$ everytime that I have a sequence $f_n$ decreasing to $0$.]

`$S = \{\lambda F(f) : \lambda \in \mathbb{R}\}$`

. Then $\lambda f \mapsto \lambda$ then extends to a nontrivial (since $F(f) \ge 1$) continuous linear functional dominated by $F$. $\endgroup$