prove $$\mathbf{E}\left[f\left(\beta_{t+h}\right)|\mathcal{F}_{t}^{0}\right]\mathbb{I}_{\left\{ \tau>t\right\} }=\mathbf{E}\left[f\left(\beta_{t+h}\right)|\beta_{t}\right]\mathbb{I}_{\left\{ \tau>t\right\} },\;\mathbf{P}\textrm{-a.s.}$$ Both sides being $\mathcal{F}_{t}^{P}$-measurable, it suffices to verify that for all $A\in\mathcal{F}_{t}^{0}$ we have $$\int_{A\cap\left\{ \tau>t\right\} }f\left(\beta_{t+h}\right)d\mathbf{P}=\int_{A\cap\left\{ \tau>t\right\} }\mathbf{E}\left[f\left(\beta_{t+h}\right)|\beta_{t}\right]d\mathbf{P}.\label{eq:equivalentMK1}$$ We observe that $\mathcal{F}_{t}^{0}$ is generated by $$\beta_{t_{n}},\ \xi_n:=\frac{\beta_{t_{n}}}{t_{n}}-\frac{\beta_{t_{n-1}}}{t_{n-1}},\ \xi_{n-1}:=\frac{\beta_{t_{n-1}}}{t_{n-1}}-\frac{\beta_{t_{n-2}}}{t_{n-2}},\ ..., \ \xi_1:=\frac{\beta_{t_{1}}}{t_{1}}-\frac{\beta_{t_{0}}}{t_{0}}\,,$$ $0<t_{0}<t_{1}<...<t_{n}=t$, for $n$ running through $\mathbb{N}$. By the monotone class theorem (see, e.g., [@key-7 I.19, I.21]) it is sufficient to prove (\[eq:equivalentMK1\]) for sets $A$ of the form $A=\{ \beta_{t}\in B,\,\xi_{1}\in B_{1},\ldots,\,\xi_{n}\in B_{n}\}$ with $B, B_{1}, B_{2},\ldots, B_{n}\in \mathcal{B}(\mathbb{R}),\, n\geq1$. Let $g:=\mathbb{I}_{B}$ and $L:=\mathbb{I}_{B_{1}\times B_{2}\times\cdots\times B_{n}}$. Then we have the equality $\mathbb{I}_{A}\!=\!g(\beta_{t})L(\xi_{1},\ldots,\,\xi_{n})$ and, setting $$\eta_{k}:=\frac{W_{t_{k}}}{t_{k}}-\frac{W_{t_{k-1}}}{t_{k-1}},\quad k=1,\ldots, n,$$ we have $\xi_{k}=\eta_{k}$ on $\{ \tau>t\}$, $k=1, \ldots, n$. But, for $r>t$, the random vector $(\eta_{1},\ldots, \eta_{n},\,\beta_{t}^{r},\beta_{t+h}^{r})$ is Gaussian and, denoting by $\textrm{cov}(X,Y)$ the covariance between two random variables $X$ and $Y$, we have that $\textrm{cov}(\eta_{k},\beta_{t}^{r})=\textrm{cov}(\eta_{k},\beta_{t+h}^{r})=0,\; k=1, \ldots, n$. Thus $(\eta_{1},...,\eta_{n})$ is independent of $(\beta_{t}^{r},\beta_{t+h}^{r})$ and, with the notation $H(x,y):=f(x)g(y)$, we also have that $L(\eta_{1},...,\eta_{n})$ is independent of $H(\beta_{t+h}^{r},\beta_{t}^{r})$. Now we can state the following lemma which will allow to complete the proof of Theorem \[TEO:The-information-process\].

\[lem:The-random-variables\] Let $H:\, \mathbb{R}^2\mapsto\mathbb{R}$ be a measurable function. Suppose that $H$ is nonnegative or such that $\mathbf{E}\left[\left| H\left(\beta_{t+h},\beta_{t}\right)\right|\right]<+\infty$. Then the random variables $H(\beta_{t+h},\beta_{t})\mathbb{I}_{\{ \tau>t\} }$ and $L(\eta_{1},...,\eta_{n})$ are uncorrelated: 