Hi everyone,

I have a question which I am quite stumped on. Consider the linear functional $l(f) = f(0)$ defined on $C([-1,1])$. By Hahn-Banach this linear functional can be extended to one on all of $L^{\infty}([-1,1])$. Now the space $(L^{\infty})^*$ is the set of all finitely additive measures which are *absolutely continuous with respect to Lebesgue*. Therefore $l$ must be a finitely additive measure $<< dx$ on $[0,1]$.

I apparently do not understand what this means for finitely additive measures since this element of $(L^{\infty})^*$ does not appear to be absolutely continuous; it is just dirac measure. Can someone help clarify this apparent inconsistency? Are the finitely additive functionals only defined on intervals $[a,b)$ or something of this nature?

Best, Dorian

notthe dirac delta measure, because evaluating $L^\infty$ functions at a point doesn't make sense. Rather, it's some weird Hahn-Banach extension of a dirac delta measure. So, while I think this looks odd to start with, the more I think about it, the less I see a contradiction: remember thatallyour finitely additive measure has to do is integrate against acontinuousfunction to evaluation at 0. Sorry, maybe that's not a very good answer, hence why it's just a comment. – Matthew Daws Sep 4 '10 at 16:45