Positive functional representation on Lp
= Positive functional representation on Lp
On a finite measure space, a positive functional $\Lambda\in(L^p)^*$ defines the finite measure $\nu(E)=\Lambda(\mathbf1_E)$. The <Radon-Nikodym theorem> gives $d\nu=\omega\,d\mu$ with $\omega\geq0$. The bound
$$
\int|f|\omega=\Lambda(|f|)
\leq\|\Lambda\|\,\|f\|_p
$$
and $L^p$-$L^q$ norm duality imply $\omega\in L^q$, after which density extends the integral representation to every $f\in L^p$.