*bridge of length $r$ associated with $W$*. If $W$ is a Brownian motion on the probability space $(\Omega,\mathcal{F},\mathbf{P})$, then the process $\beta^{r}$ is called *Brownian bridge of length $r$*.

We have the following fact concerning the measurability of the process $\beta^{r}$.

The map $(r,t,\omega)\mapsto\beta_{t}^{r}(\omega)$ of $((0,+\infty)\times\mathbb{R}_{+}\times\Omega,\,\mathcal{B}((0,+\infty))\otimes\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{F})$ into $(\mathbb{R},\,\mathcal{B}(\mathbb{R}))$ is measurable. In particular, the $t$-section of $(r,t,\omega)\rightarrow\beta_{t}^{r}(\omega)$: $(r,\omega)\mapsto\beta_{t}^{r}(\omega)$ is measurable with respect to the $\sigma$-algebra $\mathcal{B}((0,+\infty))\otimes\mathcal{F}$, for all $t\geq0$.

From the definition of $\beta^{r}$ we have that (i) the map $\omega\rightarrow\beta_{t}^{r}(\omega)$ is measurable for all $r>0$ and $t\geq0$ and (ii) the map $(r,t)\rightarrow\beta_{t}^{r}(\omega)$ is continuous for all $\omega\in\Omega$. It now suffices to proceed with the discretization of the parameter $(r,t)$ in order to define piecewise constant and measurable functions converging pointwise to $(r,t,\omega)\mapsto\beta_{t}^{r}(\omega)$ and to use standard results on the passage to the limit of sequences of measurable functions. The assertion of the lemma then follows immediately.

\[cor:measyBetaR\] The map $(r,\omega)\rightarrow\beta_{\cdot}^{r}(\omega)$ of $((0,+\infty)\times\Omega,\,\mathcal{B}((0,+\infty))\otimes\mathcal{F})$ into $(C,\,\mathcal{C})$ is measurable.

Note that $\beta^{r}$ is just as the $r$-section of the map $(r,t,\omega)\rightarrow\beta_{t}^{r}(\omega)$.

If the process $W$ is a Brownian motion on $(\Omega,\mathcal{F},\mathbf{P})$, the process $\beta^{r}$ is just a Brownian bridge which is identically equal to 0 on the time interval $[r,\,+\infty)$. If we denote 