Convergence of Radon Nikodym derivatives

Question

I apologise in advance if my question is too basic.

Some notation:

1. $(X,\cal{X})$ denotes a measurable metric space where $X$ is a metric space and $\cal{X}$ is the associated Borel sigma algebra.

2. $B(X)$ is the space of all bounded continuous functions defined on $X$.

Let $\{\mu_n\}$ and $\{\nu_n\}$ be sequences of probability measures on the above measurable space $(X, \mathcal{X})$. Assume that each $\mu_n$ is absolutely continuous with respect to $\nu_n$, with an density $h_n\in B(X)$. Suppose that $\mu_n\to \mu$, $\nu_n\to \nu$ in the weak star topology and $h_n$ converges to a bounded continuous function $h$.

Question: I would like to know if $\mu\ll\nu$. If so, is $h$ the density? If not, is there some condition in order to have $\mu\ll\nu$?

Other information that can be useful is that each $\nu_n$ and $\mu_n$ has support in a compact subset $K_n\subset X$ which increase to $X$, i.e, $X=\bigcup K_n$.

Edit: $h_n$ converges to $h$ uniformly in compacts sets

Answer 1

I'm going to assume that your space is locally compact (as well as $\sigma$-compact), so that $X$ is the union of a sequence of compact sets where each lies in the interior of the next.

In this case, the answer to your question is yes.

Fix any $g \in B(X)$ with $g \geq 0$. We need $\int_X g \, d\mu_n \to \int_X gh \, d\nu$.

Fix $\varepsilon>0$. Let $M_g:=\sup_{x\in X} g(x)$ and likewise $M_h:=\sup_{x\in X} h(x)$. Let $K \subset X$ be a compact set with $K^\circ$ sufficiently large that $$ \mu(X \setminus K^\circ) < \frac{\varepsilon}{4M_g} \hspace{4mm} \textrm{and} \hspace{4mm} \nu(X \setminus K^\circ) < \frac{\varepsilon}{4M_gM_h}. $$ Let $N \in \mathbb{N}$ be such that for all $n \geq N$, $$ \mu_n(X \setminus K^\circ) < \frac{\varepsilon}{4M_g} \ , \hspace{4mm} \max_{x \in K} |h_n(x)-h(x)| < \frac{\varepsilon}{4M_g} \ , \hspace{4mm} \left| \int_{K^\circ} gh \, d\nu_n - \int_{K^\circ} gh \, d\nu \right| < \frac{\varepsilon}{4}. $$ The third statement is possible since $\nu(\partial K)<\frac{\varepsilon}{4M_gM_h}$ and so $\int_{\partial K} gh \, d\nu < \frac{\varepsilon}{4}$. (This reasoning can be seen by adapting the argument for 3,4$\Rightarrow$5 on p3 of here.)

Then for all $n \geq N$, we have that \begin{align*} \Bigg| \int_X g \, d\mu_n & - \int_X gh \, d\nu \Bigg| \\ &\leq \ \left| \int_{K^\circ} gh_n \, d\nu_n - \int_{K^\circ} gh \, d\nu \right| \ + \ \int_{X \setminus K^\circ} g \, d\mu_n \ + \ \int_{X \setminus K^\circ} gh \, d\nu \\ &< \ \left| \int_{K^\circ} gh_n \, d\nu_n - \int_{K^\circ} gh \, d\nu \right| \ + \ \frac{\varepsilon}{4M_g}M_g \ + \ \frac{\varepsilon}{4M_gM_h}M_gM_h \\ &\leq \ \left| \int_{K^\circ} g.\!(h_n - h) \, d\nu_n \right| \ + \ \left| \int_{K^\circ} gh \, d\nu_n - \int_{K^\circ} gh \, d\nu \right| \ + \ \frac{\varepsilon}{2} \\ &< \ M_g.\max_{x \in K} |h_n(x)-h(x)| \ + \ \frac{3\varepsilon}{4} \\ &< \ \varepsilon. \end{align*}