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

Question

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

Answer 1

The problem is with the concept of absolutely continuous FINITELY additive measure. Here AC just means that $\mu E = 0$ whenever the Lebesgue measure $\lambda E$ of $E$ is zero (that is, $\mu$ is a general finitely additive finite measure on the measure algebra generated by Lebesgue measure). The point is that this condition on $\mu$ does not imply that for every $\epsilon > 0$ there is $\delta > 0$ s.t. $\lambda E < \delta $ implies $\mu E < \epsilon$.

You can get a Hahn-Banach extension of $l$ by letting $l(f)$ be the limit through some free ultrafilter of $2n\int_{-1/n}^{1/n} f(t) dt$.

Answer 2

one could be indeed misled by noting that on one hand $l(f)=f(0)$ if writen like $\int f(t) \mu(dt)$ then $\mu(dt)$ would not be absolutely continuous (AC) wrt the Lebesgue measure $\lambda$ ; where we took $f$ in $C_{0}$ (continuous) ; in this cas one could think \mu as defined by $\mu({0})=1$ and $\mu(B) =0$ if $B$ does not contain $0$ ; and hence $\mu$ is indeed not absolutely continuous wrt to Lebesgue measure ; But (see Yosida P 118 on the dual of $L^{\infty})$ : $\mu$ is simply defined by $\mu(A)= l(1_{A})$ ; and the fact that $\mu$ should be AC wrt Lebesgue follows simply from : $l(f) <= ||l|| \times |f|_{\infty}$ which you apply to $f=1_{A} : |1_{A}|_{\infty} = \lambda(A)$ ; and this is satisfied by the expression of $l$ : $l(f)= \lim 2n \int_{-1/n}^{1/n}f(t)dt$ ; hence $l$ is indeed AC wrt Lebesgue measure. Note on the other hand that $\mu(\{0\})=\lim 2n \int_{-1/n}^{1/n}1_{\{0\}}dt=0$ ; Younes Adlay