fa.functional analysis - An element of $(L^{infty})^*$ which does not seem to be a finitely additive abs. cont. measure.

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

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