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I can't understand what a vector valued measure is. In particular what does it mean to write $\nu (A)$ with A Borel measurable set?

emilia
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    A definition of vector valued measures can be found here. What part of it makes you say "I can't understand..."? Look at $\nu$ as a function on the algebra of the Borel measurable sets of some topological space (actually it is even a $\sigma$-algebra) that takes values in some Banach space. – drhab Jul 14 '15 at 09:43
  • @drhab I read the Wikipedia page but I am still not convinced by the definition they give there. How can I compute the derivative with respect to a measure, for example the Lebesgue measure, if my measure $\nu$ gives me an element of a Banach space? – emilia Jul 14 '15 at 11:09
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    First of all $\mathbb R$ can be looked at as a Banach space too. If you are looking for a derivative w.r.t. the Lebesgue measure then it seems that implicitly it is accepted that $\mathbb R$ is the Banach space you are working on. A derivative of a measure w.r.t. some measure will only exist if both measures take values in the same Banach space. Btw, I am not really familiar with vector valued measures, but I see no pitfalls yet. – drhab Jul 14 '15 at 11:17
  • Thank you for the answer. Maybe you are right: the Banach space I was looking for is just $ \mathbb R^n$. – emilia Jul 14 '15 at 12:52
  • Just some notes: the notion of differentiation generalizes at least to Banach spaces. I don't understand (yet?) the theory of differentiation for Banach spaces very well myself, but it does seem (from what I do know) that it generalizes from the theory of differentiation from $\mathbb{R}^n \to \mathbb{R}^m$ (total derivatives). Also, I remember having heard once of set-valued derivatives of measures or something, I'm not sure if they generalize the Radon-Nikodym derivative or not though https://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem#Radon.E2.80.93Nikodym_derivative – Chill2Macht Apr 22 '17 at 17:49
  • I forget where I read/skimmed about it, but I believe it was in papers discussing Alexandrov's theorem about second-order differentiability of convex functions, and also the corresponding result about Lipschitz functions and first-order differentiability (Rademacher's theorem). https://en.wikipedia.org/wiki/Alexandrov_theorem -- I think maybe it came up in the context of convex functions and the generalization of Ito's Formula, namely theorem 5.5. of Cinlar, Jacod, Protter, Sharpe 1980 https://www.researchgate.net/publication/226831525_Semimartingales_and_Markov_Processes – Chill2Macht Apr 22 '17 at 18:04
  • I think it maybe had something to do with weak derivatives https://en.wikipedia.org/wiki/Weak_derivative or distributional derivatives https://en.wikipedia.org/wiki/Distribution_(mathematics)#Differentiation https://math.stackexchange.com/questions/451746/are-weak-derivatives-and-distributional-derivatives-different https://www.math.ucdavis.edu/~hunter/m218a_09/ch3A.pdf http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/weak-derivatives.pdf I forget the name of the specific term, or where I found it, but it was a very abstract notion of differentiating measures. – Chill2Macht Apr 22 '17 at 18:07

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