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As Wikipedia mentioned, there are two interpretation of "differentiation of measures":

I was wondering

  1. if they are related to each other, or unrelated concepts?
  2. if there are other concepts that can also be viewed as differentiation of measures?

Thanks and regards!

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    @Tim(regarding (3)): Sure there are many, but I don't know:) Here is a hint for isomorphism of $I=(0,1)$ and $I^2$. Write any number $a\in I$ in binary, $a=0.a_1a_2a_3\dots$, and associate to it a point in $I^2$, $(0.a_1a_3a_5\dots,\,0.a_2a_4\dots)$. That's a measure-preserving map (but you need to fight a bit with numbers that have non-unique binary expansion to make it a bijection)2011-04-09

1 Answers 1

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regarding question 1:

If $\mu$ is a measure on $\mathbb{R}^n$, with a Radon-Nikodym derivative (of the continuous part), in respect to the Lesbeuge measure h ($\int_E\ h\ dm = \mu (E)$), then at every Lebesgue point of h, the derivative of the measure, is equal to h (practically straight from the definition). Since for every integrable function almost every point is a Lebesgue point, and the Radon-Nikodym derivative is defined up to a set of measure 0, then it is indeed the case that both "derivatives" come out the same.

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    I don't know what a substantial family is, but this looks very much like the definition you referred to in the question. (The difference is using arbitrary sets rather than balls. I guess that a "substantial family" is some restriction on the volume in relation to the diameter)?2011-05-12