A well known Theorem by Riesz says that every continuous linear functional on the space $C(X)$ of all complex valued continuous functions on the compact space $X$, is given as the integral against a regular Borel measure on $X$.
A purely measurable version of this would say something like:
Theorem Template. Let $X$ be a set and let $\mathcal A$ be a $σ$-algebra of subsets of $X$. Denoting by $\mathcal M_+(X,\mathcal A)$ the set of all $\mathcal A$-measurable functions from $X$ to $[0,+\infty]$ (extended positive reals), suppose we are given an additive, positively homogeneous map $$I:\mathcal M_+(X,\mathcal A)\to[0,+\infty], $$ satisfying [blank]. Then there exists a positive measure $\mu$ on $(X,\mathcal A)$, such that $$I(f)=\int_Xf(x)d\mu(x),$$ for all $f$ in $\mathcal M_+(X,\mathcal A)$.
What would be a minimal set of axioms to put in place of [blank]?
I strongly suspect that this must be somewhere in the literature.
Is there a reference for a result as above?
