I have a question regarding Proposition 3.12 of Chapter 2 of "Ordered Topological Vector Spaces" by Peressini, which states the following:
Suppose that $E$ is a locally convex space ordered by a cone $K$ with a nonempty interior. If $\{y_{\alpha} :\alpha \in I\}$ is a net in the dual cone $K'$ of $K$ in $E'$ that converges to $\theta$ for $\sigma(E', E)$, then $\{y_{\alpha} :\alpha \in I\}$ converges to $\theta$ for $\beta(E', E)$.
Above $\sigma(E', E)$ stands for the weak topology on the dual $E'$ and $\beta(E', E)$ stands for the strong topology on $E'$.
I have some doubts about this theorem because when I try to apply it to Banach spaces of bounded measurable functions I seem to get conclusions that are clearly not true.
Here is an example. Let $(\Omega, \Sigma)$ be a measurable space, write $B(\Sigma)$ for the Banach space of the real valued bounded measurable functions on $(\Omega, \Sigma)$ and order it with respect to the cone of the non-negative functions (this cone certainly has non-empty interior: for example the constant function 1 is an interior point.)
Now let $\mu_{i}$ be a sequence of probability measures on $(\Omega, \Sigma)$ converging setwise to a measure $\mu$, and let $L_{i}$ and $L$ be the corresponding positive linear functionals on $B(\Sigma)$ given by integration. Then $L_{i}$ converges to $L$ pointwise on simple functions and hence by density and uniform boundedness on all functions in $B(\Sigma)$. So by the theorem above $L_{i}$ converges to $L$ strongly, that is, in the norm of the dual of $B(\Sigma)$. But that means that the measures $\mu_{i}$ converge in the (total) variation norm. So we get that setwise convergence implies convergence in total variation, which is known to be false.
So my question is: what is wrong with the argument above? Am I misunderstanding the theorem from Peressini's book or is there something wrong with its statement?