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Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have that the Radon-Nikodým derivatives $h_n$ of $\nu_n$ wrt. $\mu_n$ converge to the Radon-Nikodým derivative $h$ of $\nu$ wrt. $\mu$?

Does the situation becomes easier if we assume that $0\leq h_n \leq 1$ and $0\leq h\leq 1$?

Does the results carry over if we consider finite random measures instead of deterministic measures? And is the result still available if the convergence is only in the distributional sense, i.e. for compactly supported smooth functions?

Thanks a lot for your help!

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  • $\begingroup$ The standard example to look at here is the system of Haar functions where things go wrong—these are generally defined on the unit interval but this has no relevance for your query. By the way, your objects, as measures, can be regarded as Schwartzian distributions and then everything is fine except that the derivatives have to be taken in the distributional sense which might not correspond to your requirements. $\endgroup$
    – user131781
    Commented Jul 5, 2020 at 10:10
  • $\begingroup$ @user131781 What do mean when you say that everything is fine if I consider them as Schwartzian distributions? I have never heard of taking Radon-Nikodým derivatives in the distributional sense... Do you have a reference? This is perhaps exactly what I need :) $\endgroup$
    – Mushu Nrek
    Commented Jul 6, 2020 at 14:37

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The answer is negative for a.e. pointwise convergence as well as $L^1$ convergence with respect to $\mu$. Let $\mu_n=\mu$ be half the Lebesgue measure, color the torus as a checkerboard with $2^n$ cells, and assume $h_n$ takes value $1$ on black cells and $0$ on white cells. Then $\nu_n=h_n\mu \to \mu$ $*$-weakly, but for all $n$ the function $h_n-h$ is everywhere $\pm\frac12$.

To control densities, you want convergence in total variation (or variations on this).

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Mushu Nrek
    Commented Jul 6, 2020 at 14:45

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