in ${\widetilde}{\cF}_{\infty}^{Y}$ of the form $$\begin{aligned}
C=\left\{\left(Y_{u_{1}},\ldots,Y_{u_{m}}\right)\in B\right\},\quad 0\leq u_{1}<u_{2}<\ldots<u_{m},\quad B\subseteq\mathbf{E}^{m},\quad m\in\bN.\end{aligned}$$ Clearly, $\cC$ is an algebra.

We first show that for any $C\in \cC$ the limit in exists and that an explicit formula for it can be derived. In fact, Lemma \[lem:ConstBarPiCylSet\] shows that the limit in exists, and belongs to $[0,1]$, for all the cylinder sets of the form . Thus, for a cylinder set $C\in \cC$ an explicit formula for the limit on the right-hand side of can be obtained as follows. First, we refine the partition $0\leq u_{1}<u_{2}<\ldots<u_{m}$ so that each subinterval of the partition $0<s_{1}<\ldots<s_{n}$ contains at least one of the $u_i$’s. Clearly, since $B_m$ is finite, $A$ can be decomposed into a finite union of disjoint cylinder sets of the form on the refined partition. Moreover, provides an explicit formula for the limit in for each of those cylinder sets of the form on the refined partition. Finally, taking the finite sum over all those limits, we obtain the limit in for $C$. In particular, for every cylinder set $C$, the limit in is Borel measurable with respect to $(s_{1},\ldots,s_{n})$ in $\Delta_{n}$.

In the second step we will demonstrate that the limit in can 