I am struggling with two versions of the Cameron–Martin Theorem.
1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto f(t), t\geq 0, f\in C[0,1])$ is the sigma Algebra generated by projection maps and $\mathbb{L}_0$ the distribution of the standard Brownian Motion $B_t:(\Omega,\mathcal{F})\to (C[0,1],\mathcal{C})$. Analogously, we define for $F:[0,\infty)\to\mathbb{R}$ the distribution $\mathbb{L}_F=\mathcal{L}_P(B_t+F(t), t\geq 0)$. Let's right away assume that $D[0,1]\ni F(s)=\mu s$ with $D[0,1]$ the Dirichlet space.
Peres and Moerters show that $\mathbb{L}_F\sim\mathbb{F}_0$ and $\frac{d\mathbb{L}_F}{d\mathbb{L}_0}(B)=\exp{(-\frac{1}{2}\mu^2 t+\frac{t}{2}\mu B(t))}=:Z_{\mu}(t)$.
2) Steve Lalley from University of Chicago on the other hand shows that $Z_{\mu}(t)$ is a positive martingale wrt the Brownian Filtration $(\mathcal{F}_t){t\geq 0}$ and $E_P[Z_{\mu}(t)]=1$. It defines a new probability measure
$P_\mu(A):=E_P[Z_\mu(t)\mathbf{1}_A]$
where $A\in\mathcal{F}_t$. His version of the Cameron–Martin Theorem states that under $P_\mu$ the process $(B_t)_{t\geq 0}$ has the same distribution as the drifted Brownian motion $(B_t+\mu t)_{t\geq 0}$ unter $P_0=P$. I dont see how versions 1) and 2) are equivalent.
I am very grateful for any help!
nbt
Let $B^\mu$ be the drifted Brownian motion. The first formulation says that $$ P(B^\mu\in A) = \int_A Z_\mu d\mathbb L_0; $$ the second, that $P_\mu(B\in A) = P(B^\mu\in A)$. However, $P_\mu(B\in A) = P^\mu (A) = \int_A Z_\mu d\mathbb L_0$ according to its definition. So indeed two formulations are the same.