The "fundamental theorem of duality" states:
If $X$ is a real linear space and $f, f_1,...,f_n$ are linear functionals on $X$, then $f$ lies in the span of $f_1,...,f_n$ (i.e. $f = \sum_{i=1}^n \lambda_i f_i$, where $\lambda_i \in \mathbb R$) if and only if $\bigcap_{i=1}^n ker \ f_i \subseteq ker \ f$ (where $ker \ g = \{x \in X: g(x)=0\}$).
My question is:
Can someone please direct me to a reference that generalizes this result to arbitrary collections of linear functionals?
In other words, if $I$ is an arbitrary index set, I am wondering how to characterize $$\bigcap_{i \in I} ker \ f_i \subseteq ker \ f.$$ Intuitively, I suspect that the equivalent statement is something like: there exists a finitely additive signed measure $\mu$ on $2^I$ such that for all $x \in X$ $$ f (x) = \int f_i (x) \ \mu(di).$$ But this will be tricky as it involves finitely additive integration of potentially unbounded functions. (Clearly, though, this condition is sufficient for the condition about kernels to hold. The difficulty is in showing that it is also necessary.)
Anyway, surely generalizations of the fundamental theorem of duality have been explored, and any references or pointers would be appreciated.