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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

  1. $\forall$ $\varepsilon>0$ $\exists$ $\delta>0$ such that $\nu(A)<\delta$ $\Rightarrow$ $\mu(A)<\varepsilon$
  2. 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

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  • $\begingroup$ Isn't it just to take some basis and study the positive and negative parts of the different coefficients? That is, let $\{b_\alpha\}$ be a basis of the Banach space and define $\mu_\alpha$ by $\mu = \sum_\alpha \mu_\alpha b_\alpha$, and then $\mu_\alpha^\pm$ to be the positive and negative parts of $\mu_\alpha$. Ditto for $\nu$. Then apply the above for $\mu_\alpha^+ \ll \nu_\alpha^+$ and $\mu_\alpha^- \ll \nu_\alpha^-$. $\endgroup$
    – md2perpe
    Oct 27, 2017 at 19:45

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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.

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