Let $(X,\mu)$ be a measure space and $\mu(X)< + \infty$, $\phi$ be a bounded linear functor on $L^1(\mu)$. Prove that there exists a positive measure $\lambda$ on $X$ such that $\phi(f) = \int_X f d\lambda$ for any $f \in L^1(\mu)$.
This appears like Riesz representation theorem to me. But I don't know how to do it.
Thank you very much.