Dears,
the question is - why there are finitely-additive measures in dual to L infinity, not only sigma-additive? I was thinking, that sigma-additive measures are related with embedded elements of L1 into (Linfinity)*, whereas finitely-additive [and not sigma-additive] are related with "the rest". Can someone explain that or give me link to article or pdf where it is well explained? I want to know what gives that difference.
I think the best think to do is to have an example of a functional in the dual of $L_\infty$ of some space (say the Borel space $\big(\mathbb{N}, 2^{\mathbb{N}}, \#\big)$ that is not a $\sigma$--finite measure. A classical example is to start with $\mathcal{c}$, the space of convergent sequences and define $\Lambda\phi:=\lim_n\phi(n)$. This is a continuous linear functional on $\mathcal{c}$. Using Hahn-Banach theorem, $\Lambda$ can be extended to the whole of $\ell_\infty$. Convince yourself that $\Lambda$ is not a $\sigma$--additive measure. It also follows from this that there is no $f\in\ell_1$ for which $\Lambda\phi=\int \phi f_1\,d\mu$ whenever $\phi\in\ell_\infty$.
In general, the finitely additive measures in a measurable space $(X,\mathscr{F})$ are go under the name of charges. There is also a notion of variation $|\nu|$ for a charge $\nu$. The charges in $L_\infty(X,\mathscr{F},\mu)$ also have the property that they are absolutely continuous with respect to the ambient measure $\mu$.
A modern and somewhat simple exposition of this things are in Aliprantis' Infinite dimensional Analysis, Hitchhiker's guide, chapter 14.
All this depends on the axiom of choice.