be extended to a probability measure on $\sigma(\cC)={\widetilde}{\cF}_{\infty}^{Y}$. We start from verifying the countable additivity of $\overline{\bP}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ on $\cC$ for any fixed $0<s_{1}<\ldots<s_{n}$.

Since $\mathbf{E}$ is a finite set, if $(C_{k})_{k\in\bN}$ is a sequence of disjoint cylinder sets in $\cC$ such that their union also belongs to $\cC$, then only finite many of them are non-empty. Therefore, it suffices to verify the finite additivity of $\overline{\bP}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ on $\cC$. Let $C_{1},\ldots,C_{k}\in\cC$ be disjoint cylinder sets, then there exists $m\in\mathbb{N}$ and $0\leq u_{1}<u_{2}<\ldots<u_{m}$, such that $$\begin{aligned}
C_{\ell}=\left\{\left(Y_{u_{1}},\ldots,Y_{u_{m}}\right)\in B_{\ell}\right\}\,\,\,\,\text{for some }\,B_{\ell}\subseteq\mathbf{E}^{m},\quad\ell=1,\ldots,k.\end{aligned}$$ Each $\overline{\bP}^{i}(C_{\ell}\,;s_{1},\ldots,s_{n})$ can be represented as $$\begin{aligned}
\overline{\bP}^{i}(C_{\ell}\,;s_{1},\ldots,s_{n})=\sum_{A_{\ell}\in\cC_{\ell}}\overline{\bP}^{i}(A_{\ell}\,;s_{1},\ldots,s_{n}),\quad j=1,\ldots,k,\end{aligned}$$ where $\cC_{\ell}$, $\ell=1,\ldots,k$, are disjoint classes of disjoint simple cylinder sets. Therefore, we have $$\begin{aligned}
\sum_{\ell=1}^{k}\overline{\bP}^{i}(C_{\ell}\,;s_{1},\ldots,s_{n})&=\sum_{\ell=1}^{k}\sum_{A_{\ell}\in\cC_{\ell}}\overline{\bP}^{i}(A_{\ell}\,;s_{1},\ldots,s_{n})\\
&=\sum_{A\in\cC_{1}\cup\cdots\cup\cC_{k}}\overline{\bP}^{i}(A\,;s_{1},\ldots,s_{n})=\overline{\bP}^{i}\left(\bigcup_{\ell=1}^{k}C_{\ell}\,;s_{1},\ldots,s_{n}\right).\end{aligned}$$ Note that for any $0<s_{1}<\ldots<s_{n}$, $\overline{\bP}^{i}(C\,;s_{1},\ldots,s_{n})\leq 1$ for all $C\in\cC$. By the Carathéodory extension theorem, for any $0<s_{1}<\ldots<s_{n}$, $\overline{\bP}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ can be uniquely extended to a probability measure on $({\widetilde}{\Omega},{\widetilde}{\cF}_{\infty}^{Y})$.

Let $\Delta_{n}:=\{(s_{1},\ldots,s_{n})\in\bR^{n}\,|\,0<s_{1}<\ldots<s_{n}\}$ and $$\begin{aligned}
\cD_{1}:=\left\{\left.A\in{\widetilde}{\cF}_{\infty}^{Y}\,\right|\overline{\bP}^{i}(A\,;\,\cdot,\cdots,\cdot)\,\,\text{is Borel measurable on }\Delta_{n}\right\}.\end{aligned}$$ We will show that $\cD_{1}={\widetilde}{\cF}_{\infty}^{Y}$. Towards this end, we first observe that and imply that, for any $A\in \cC$, $\overline{\bP}^{i}(A\,;\,\cdot,\cdots,\cdot)$ is Borel measurable with respect to $(s_{1},\ldots,s_{n})$ on $\Delta_{n}$, and thus $\cD_{1}\supset\cC$.

Next, we will show that $\cD_{1}$ is a monotone class. For this, let $(A_{k})_{k\in\mathbb{N}}\subset\cD_{1}$ be an 