Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a positive functional, that is $Lf \ge 0$ if $0 \le f \in M$ (note that this condition can be void if $M$ does not contain non-trivial positive functions) , there are conditions under which $L$ can be extended as a positive functional defined on $X$. These results go back to Bauer, Namioka, Kantorovich and can be found in the book of H. Schaefer, Topological vector spaces. Counteraxmples are easily constructed by choosing positive functionals discontinuous on $M$; they cannot be extended to positive functionals on $X$, since positivity on the whole space implies continuity. An example is integration against $1/x$ wich is a positive functional on bounded functions in $(0,1)$, with support far from the origin, but cannot extended to $L^p(0,1)$.
The question I have is the following: I do not know examples of positive and continuous functionals on subspaces which cannot be extended as positive functionals on $X$. I recall that I am using the natural order. They should exist, and I hope that somebody knows.
