Extension of Cameron-Martin formula via monotone class theorem

Question

my question revolves around the Cameron-Martin theorem:

Let $(\mathcal{C}_{(0)}[0,1],\mathcal{B}(\mathcal{C}_{(0)}),\mu)$ be the Wiener space (i.e. continuous functions starting in $0$, equipped with the law of a Brownian motion $B:(\Omega,\mathcal{A},P)\to\mathcal{C}_{(0)}[0,1]$, $\mu:=B[P]$ and $f_0\in\mathcal{C}_{(0)}[0,1]$ an element in the Cameron-Martin space, i.e. $f_0$ is absolutely continuous and its derivative is in $L^2[0,1]$. Then the image measure $S[\mu]$ under the shift $S(f)=f+f_0$ is absolutely continuous with respect to $\mu$ and for every bounded, measurable functional $F:\mathcal{C}_{(0)}\to\mathbb{R}$ we have $\int_{\mathcal{C}_{(0)}} F(f)dS[\mu]=\int_{\mathcal{C}_{(0)}} F(f)\frac{dS[\mu]}{d\mu}(f)d\mu(f)$ $\quad\quad(*)$

where the density is given by

$\frac{dS[\mu]}{d\mu}(f)=\exp\bigl(-\frac{1}{2}\int_0^1|f'_0(t)|^2dt+\int_0^1f'_0(t)df(t)\bigr)$

In the book "Brownian Motion" by Schilling and Partzsch the proof was completed for continuous $F$ (notation as above) and then they said "Using standard approximation arguments from measure theory, the formula $(*)$ is easily extended from continuous $F$ to all bounded measurable $F$.

Now, I wonder if thats possible to show via monotone class theorem:

1. Let $M:=\{F:\mathcal{C}_{(0)}\to\mathbb{R}|\ (*) \text{ holds}\}$ be a linear subspace of $\mathbb{R}^{\mathcal{C}_{(0)}}$.

2. Assume we know that the generating system $\mathcal{E}:=\sigma(\bigcup_{t\in[0,1]}\pi_t^{-1}(\mathcal{B}(\mathbb{R}))\ni\mathcal{C}_{(0)}[0,1]$ of the Wiener $\sigma$-algebra $\mathcal{B}(\mathcal{C}_{(0)})$ is a $\pi$-system, i.e. $\cap$-stable. We denoted $\pi_t$ the projection $f\mapsto f(t)$.

3. For any $A\in\mathcal{E}$, $\mathbf{1}_{A}\in M$. Indeed, $\int_{\mathcal{C}_{(0)}}\mathbf{1}_A(f)dS[\mu]=S[\mu](A)$ and $\int_{\mathcal{C}_{(0)}}\mathbf{1}(S(f))d\mu(f)=\int_{\mathcal{C}_{(0)}}(\mathbf{1}_A\circ S)d\mu=\int_{\mathcal{C}_{(0)}}\mathbf{1}_A(f)dS[\mu]=S[\mu](A)$.

   4. For any increasing sequence of bounded, non-negative functions $(F_n)_{n\in\mathbb{N}}\subset M$, with $F_n\longrightarrow F$ pointwise, we immediately follow from monotone convergence that $F\in M$.

Hence the space of all bounded, measurable functions is in $M$ as well.

If the argument above is fine, how can I show that in 2. the generating system $\mathcal{E}$ is a $\pi$-system?

Thanks for your help!