0}$$

The map $\beta:\,(\Omega,\,\mathcal{F})\rightarrow(C,\,\mathcal{C})$ is measurable.

The map $(r,\omega)\rightarrow\beta^{r}(\omega)$ of $((0,+\infty)\times\Omega,\,\mathcal{B}((0,+ \infty))\otimes\mathcal{F})$ into $(C,\,\mathcal{C})$ is measurable because of Corollary \[cor:measyBetaR\]. By definition, the map $\omega\rightarrow(\tau(\omega),\omega)$ of $(\Omega,\mathcal{F})$ into the measurable space $((0,+\infty)\times\Omega,\,\mathcal{B}((0,+\infty))\otimes\mathcal{F})$ is measurable. The statement of the lemma follows from the measurability of the composition of measurable functions.

We now consider the conditional law with respect to $\tau$. As common, by $\mathbf{P}_\tau$ we denote the distribution of $\tau$ with respect to $\mathbf{P}$.

\[lem:If–is\]If $\tau$ is independent of the Brownian motion $W$, then for any measurable function $G$ on $\left(\left(0,+\infty\right)\times C,\mathcal{B}\left(\left(0,+\infty\right)\right)\otimes\mathcal{C}\right)$ such that $G(\tau,\, W)$ is integrable it follows that $$\mathbf{E}\left[G\left(\tau,\, W\right)|\sigma\left(\tau\right)\right]\left(\omega\right)=\left(\mathbf{E}\left[G\left(r,W\right)\right]\right)_{r=\tau\left(\omega\right)},\;\mathbf{P}\textrm{-a.s.}$$

See, e.g., [@key-8 Ch. II.7, p. 221].

\[cor:LEM if–is\] If $h:\,\left((0,+\infty)\times C,\mathcal{B}\left(\left(0,+\infty\right)\right)\otimes\mathcal{C}\right)\mapsto\left(\mathbb{R},\mathcal{B}(\mathbb{R})\right)$ is a measurable function such that $\mathbf{E}\left[|h\left(\tau,\,\beta\right)|\right]<+\infty$, then $$\mathbf{E}\left[h\left(\tau,\,\beta\right)|\tau=r\right]=\mathbf{E}\left[h\left(r,\,\beta^{r}\right)\right],\; \mathbf{P}_{\tau}\textrm{-a.s.},$$ or, equivalently, $$\mathbf{E}\left[h\left(\tau,\,\beta\right)|\sigma\left(\tau\right)\right]\left(\omega\right)=\left(\mathbf{E}\left[h\left(r,\,\beta^{r}\right)\right]\right)_{r=\tau\left(\omega\right)},\; \mathbf{P}\textrm{-a.s.}$$

The last two formulas provide a useful connection between the law of Brownian bridges and the conditional law with respect to $\sigma(\tau)$ of a generic functional involving $\tau$, the Brownian motion $W$ and the map $\beta$ defined in (\[eq:beta def 0\]).

\[sec:Definition-and-First\] Definition and First Properties
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Let $W=(W_{t},\, t\geq0)$ be a standard Brownian motion (with respect to its completed natural filtration $\mathbb{F}^{W}=(\mathcal{F}_{t}^{W})_{t\geq0}$) starting from 0. Let $\tau:\,\Omega\rightarrow(0,\infty)$ be a strictly positive 