I am looking for the following statemente. 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 assumme a lot of regularity on $X$ i.e. it is compact, locally compact... Also we can assumme some regularity on $\mu$, $\nu$ i.e. atom-freeness the support is X...

It 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 $U_i$?

Thanks

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 $U_i$?

Thanks