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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.

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    This is the abstract version of the Riesz representation theorem. Check out [this wikibook entry](http://en.wikibooks.org/wiki/Measure_Theory/Riesz%27_representation_theorem) for a proof on $C_c(X)$ (which is dense in $L^1(X,d\mu)$) or for the full picture see Rudin "Real and Complex Analysis" or Lax "Functional Analysis"2012-11-22
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    @icurays1 What happens if there is not a topological structure on $X$?2012-11-23

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