Let $F$ by a process adapted to the filtration of a standard Brownian motion. Suppose that the Doleans Dade martingale exists and is a martingale. $F$ is a measure on $W_0^{1,2}$, call it $\nu$. Let $\mu_0$ be the standard Wiener measure on $C[0,T]$ and by Girsanov let $\mu^\ast=M(T)\mu_0$ be the measure on $C[0,T]$ given by Girsanov. Define the map
$$f:C[0,T]\times W_0^{1,2}[0,T]\to C[0,T]$$
by $$f(\phi,\psi)=\phi+\psi. $$
Then is the pushforward measure $\mu$ defined by
$$\mu(A):=(\mu_0,\nu)(f^{-1}(A))$$
for Borel $A\subset C[0,T]$ the measure $$\mu=\mu^\ast?$$
This seems very reasonable, I would like a reference if possible, or disproof.
