Let $\mu\_0$ be the standard Wiener measure on $C[0,T]$. Let $\mu\_n$ be a sequence of measures with $\mu\_n\ll \mu\_0$ for all $n$ and so that the weak$^\ast$ limit of $\mu\_n$ exists, call it $\mu$. Is it true that $\mu \ll \mu\_0$?


I know for general measures this is not true. For example we can have a Gaussian with variance $\varepsilon$ and send $\varepsilon \to 0$.


But what about for Girsanov measures?



Is the weak* limit of Girsanov measures also a Girsanov measure?
This goes as wrong as can be and has nothing to do with any property of Wiener measure.


**Theorem:** Let $X$ be a separable metric space and $\nu$ a measure on $X$. The set of measures absolutely continuous with respect to $\nu$ is dense in the space of measures supported on the support of $\nu$ in the topology of weak convergence of measures.


Proof: Assume without loss of generality that the support of $\mu$ is all of $X$. It is known that the set of measures with finite support is dense, so it suffices to approximate such measures. So let $\mu$ be a Borel measure on $X$ with finite support $S$. By the definition of the topology of weak convergence, it suffices to show that there exists for every finite family $\mathcal{G}$ of bounded continuous functions on $X$ and every $\epsilon>0$ some measure $\tau$ on $X$ absolutely continuous with respect to $\nu$ such that
$$\bigg|\int g~\mathrm d\mu-\int g~\mathrm d\tau\bigg|<\epsilon$$
for all $g\in\mathcal{G}$. For each $s\in S$, let $V\_s$ be an open neighborhood on which each $g\in\mathcal{G}$ varies by less than $\epsilon/\# S$. We can and do choose the $V\_s$ to be disjoint. Since $\nu$ has full support, we have $\nu(V\_s)>0$ for all $s\in S$. Define a measurable function $h:X\to\mathbb{R}$ by letting $h(x)=\mu(s)/\nu(V\_s)$ for $x\in V\_s$ and $h(x)=0$ if $x$ is in no $V\_s$. Let $\tau$ be the measure that has Radon-Nikodym derivative $h$ with respect to $\nu$. Take any $g\in\mathcal{G}$. Then,
$$\bigg|\int g~\mathrm d\mu-\int g~\mathrm d\tau\bigg|=\bigg|\int g~\mathrm d\mu-\int hg~\mathrm d\nu\bigg|$$
$$=\bigg| \sum\_{s\in S}\bigg(g(s)\mu(s)-\int\_{V\_s}hg~\mathrm d\nu\bigg)\bigg|$$
$$\leq \sum\_{s\in S}\bigg|\bigg(g(s)\mu(s)-\int\_{V\_s}hg~\mathrm d\nu\bigg)\bigg|$$
$$<\sum\_{s\in S} \mu(s)\epsilon/\# S=\epsilon.$$


