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I am looking for the following statement. Let $X$ be a topological space and let $\mu$, $\nu$ be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by $f$ the Radon derivative.

We can assume a lot of regularity on $X$ i.e. it is compact, locally compact... Also we can assume some regularity on $\mu$, $\nu$ i.e. atom-freeness the support is X...

Is then the following statement true.

For each $x$ in $X$ outside a measure zero-set and for each sequence of nested open sets $U_i$ whose intersection is only $x$ it follows that $f(x)=\liminf \mu(U_i)/\nu(U_i) $

Are there any further assumption for $U_i$?


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By Luzin's theorem, $f$ is continuous except for a set of arbitrarily small measure; if $f$ is continuous at $x$ then your limit equals $f(x)$. This has to be made more accurate though. – Yulia Kuznetsova Jan 11 '12 at 13:32
Try to search for material on Lebesgue points. If $\nu$ is the Lebesgue measure in $\mathbb R^n$, then almost every point is a Lebesgue point and so has your property. – Yulia Kuznetsova Jan 11 '12 at 14:03
@Yulia: By Luzin's theorem, the restriction of $f$ to some set $E$ of almost full measure is continuous or, if you prefer, $f$ equals to some other continuous function $g$ outside a set of small measure. This is very different from the continuity of $f$ itself. The theorem on Lebesgue points also requires very particular shapes of $U_i$ to employ covering lemmas. @Klaus The desired property is currently stated so sloppily that it doesn't hold even at points of continuity: take $X=\mathbb R$, $x=0$, and $U_i=(-1/i,1/i)\cup (i,+\infty)$. – fedja Jan 11 '12 at 15:22
@yulia: many thank's, embarassingly I did not know the notion of Lebesgue points. As X is also metric and one measure is the Hausdorff measure there might be some theorem that almost every point is lebesgue. @fedja: Many thank's for correcting me, my question is missleading. Actually, I am looking for a criterion for the shape of $U_i$. – Klaus Jan 11 '12 at 15:44
@fedja: thank you, I forgot much of it since using. @Klaus: I remember reading a good review close to this topic: A. Bruckner, “Differentiation of integrals,” Amer. Math. Monthly 78 (9, part II) (1971). – Yulia Kuznetsova Jan 11 '12 at 16:11

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