Absolute continuity for vector measures By definition, a real measure $\mu$ is absolutely continuous with respect to a real measure $\nu$ if, whenever $\nu(A)=0$, then $\mu(A)=0$. We also know that, for finite real measures, absolute continuity is equivalent to every of the following conditions


*

*$\forall$ $\varepsilon>0$ $\exists$ $\delta>0$ such that $\nu(A)<\delta$ $\Rightarrow$ $\mu(A)<\varepsilon$

*There exists a $\nu$-integrable real function $f$ such that
$$\mu(A)=\int_A f d\nu$$


Does anyone know whether there exist similar conditions for vector measures, i.e. measures taking values in Banach spaces? At least, can you suggest any book where you think I may find more information?
Thanks,
Alessandro
 A: The Radon-Nikodym theorem (i.e. the theorem that says those two conditions are equivalent for $\sigma$-finite measures) fails to hold in general for the Bochner integral (the integral with respect to a Banach space valued measure). You can find an easy counterexample for this in the exercises of "Measure Theory" By Cohn. In short it goes as follows.
Consider the Lebesgue measure $\lambda$ define on the Borel subsets of $[0,1]$. Then you define the Banach space valued measure by
$$
\nu: \mathcal{B}([0,1]) \to L^1([0,1], \lambda): A \mapsto \chi_A.
$$
You can prove that $\nu$ is absolutely continuous with respect to $\lambda$, but there is no function $f: [0,1] \to L^2([0,1], \lambda)$ such that 
$$
\nu(A) = \int \chi_Af d\lambda
$$
holds.
If you impose some extra conditions on the Banach space $X$ in question, the Radon-Nikodym theorem might hold for all vector measures that take values in this Banach space. We say that such Banach spaces hold the Radon-Nikodym property.
Examples of these conditions are "$X$ is a seperable dual space" or "$X$ is reflexive" (note that this implies that the Radon-Nikodym property holds for all Hilbert spaces). More general conditions are proven in "Vector Measures" by Diestel & Uhl. In this book pages 217-219 might also be of interest, it gives a summary of equivalent formulations of the Radon-Nikodym property and a list of particular spaces with the property. 
