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I am working with the space $C[a,b]$ continuous functions on $[a,b]$ with the maximum norm as a closed subespace of $L^\infty[a,b]$. Over this I define the bounded functionatl $\psi(f)=f(a)$ and then by the Hahn-Banach Theorem I can extend this to a continuous linear functional on $L^\infty[a,b]$.

I need to prove now that there is NO function $h \in L^1[a,b]$ such that $$\psi(f)=\int_a^b hf$$ for all $f \in L^\infty[a,b]$.

Can someone give me a hint how to do that? I was trying to use the fact that this integral has to be $f(a)$ when $f$ is continuous and use some particular $f$, but I could not conclude what I want.

Thank you all.

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    Maybe try to get a contradiction from a sequence $f_k$ of functions which have support shrinking to $a$? – 2017-01-05

1 Answers 1

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HINT For $h\in L^1[a,b]$, the measure $\mu$ defined by $$ \mu(A)=\int_Ah\ dm $$ is absolutely continuous with respect to $m$.