increasing sequence of events, so that, for any $0<s_{1}<\ldots<s_{n}$, we have $$\begin{aligned}
\overline{\bP}^{i}\left(\bigcup_{k=1}^{\infty}A_{k}\,;\,s_{1},\ldots,s_{n}\right)=\lim_{m\rightarrow\infty}\overline{\bP}^{i}\left(A_{m}\,;\,s_{1},\ldots,s_{n}\right).\end{aligned}$$ Thus, $\overline{\bP}^{i}(\cup_{k}A_{k}\,;\cdot,\cdots,\cdot)$, being a limit of a sequence of Borel measurable functions on $\Delta_{n}$, is Borel measurable on $\Delta_{n}$, and hence $\cup_{k}A_{k}\in\cD_{1}$. Similarly, one can show that if $(A_{k})_{k\in\mathbb{N}}\subset\cD_{1}$ is a decreasing sequence of events, then $\cap_{k}A_{k}\in\cD_{1}$. Therefore, $\cD_{1}$ is a monotone class, and by the monotone class theorem $\cD_{1}=\sigma(\cC)={\widetilde}{\cF}_{\infty}^{Y}$.

It remains to show that holds true. In view of and , for any cylinder set $A\in\cC$, $$\begin{aligned}
\overline{\bP}^{i}(A\,;s_{1},\ldots,s_{n})&=\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}\frac{{\widetilde}{\bP}^{i}\left(A,\,s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}{{\widetilde}{\bP}^{i}\left(s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}\\
&=\frac{\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}(\Delta s_{1}\cdots\Delta s_{n})^{-1}\,{\widetilde}{\bP}^{i}\left(A,\,s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}{\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}(\Delta s_{1}\cdots\Delta s_{n})^{-1}\,{\widetilde}{\bP}^{i}\left(s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}\\
&=\frac{\partial^{n}}{\partial s_{1}\cdots\partial s_{n}}{\widetilde}{\bP}^{i}\left(A,\,S_{k}\leq s_{k},\,k=1,\ldots,n\right)\cdot\left(\prod_{k=1}^{n}q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\right)^{-1}.\end{aligned}$$ Hence, for any $A\in\cC$, $$\begin{aligned}
&\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\overline{\bP}^{i}(A;s_{1},\ldots,s_{n})\prod_{k=1}^{n}q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\operatorname{d \!}s_{1}\cdots \operatorname{d \!}s_{n}\\
&\quad=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\frac{\partial^{n}}{\partial s_{1}\cdots\partial s_{n}}{\widetilde}{\bP}^{i}\left(A,\,S_{k}\leq s_{k},\,k=1,\ldots,n\right)\operatorname{d \!}s_{1}\cdots \operatorname{d \!}s_{n}={\widetilde}{\bP}^{i}(A),\end{aligned}$$ and thus $\cC \subset \cD_{2}$, where $\cD_{2}:=\left\{\left.A\in{\widetilde}{\cF}_{\infty}^{Y}\,\right|\text{\eqref{eq:TildePiBarPi} holds for }A\right\}$. Next, for any increasing sequence of events $(A_{k})_{k\in\bN}\subset\cD_{2}$, we have that $$\begin{aligned}
{\widetilde}{\bP}^{i}\left(\bigcup_{k=1}^{\infty}A_{k}\right)=\lim_{k\rightarrow\infty}{\widetilde}{\bP}^{i}(A_{k})&=\lim_{k\rightarrow\infty}\int_{0}^{\infty}\!\!\!\int_{s_{1}}^{\infty}\!\!\cdots\!\int_{s_{n-1}}^{\infty}\!\overline{\bP}^{i}(A_{k};s_{1},\ldots,s_{n})\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})}\operatorname{d \!}s_{1}\cdots \operatorname{d \!}s_{n}\\
&=\int_{0}^{\infty}\!\!\!\int_{s_{1}}^{\infty}\!\!\cdots\!\int_{s_{n-1}}^{\infty}\!\overline{\bP}^{i}\!\left(\bigcup_{k=1}^{\infty}A_{k};s_{1},\ldots,s_{n}\right)\!\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})}\operatorname{d \!}s_{1}\cdots \operatorname{d \!}s_{n},\end{aligned}$$ where the last equality follows from the dominated convergence theorem as well as the fact that $\overline{\bP}^{i}(A_{k};s_{1},\ldots,s_{n})\leq 1$, for all $k\in\mathbb{N}$ and $0<s_{1}<\ldots<s_{n}$. Hence, $\cup_{k}A_{k}\in\cD_{2}$. Similarly, one can show that if $(A_{k})_{k\in\bN}\subset\cD_{2}$ is a decreasing sequence, then $\cap_{k}A_{k}\in\cD_{2}$. Therefore, $\cD_{2}$ is a monotone class, and by the monotone class 