Asumme that $Y, \{X_n\}$ are random variables on a probability space $(\Omega, \mathcal{A},P)$. Assume that $Y$ is measurable with respect to $\sigma(X_1,X_2,\ldots)$. Define $\mathbb{X}=(X_1,X_2,\ldots)$. Then $\mathbb{X}$ is a random vector on $\mathcal{B}(\mathbb{R}^{\mathbb{N}})$. We also have that $Y$ is then measurable with respect to $\sigma(\mathbb{X})$. By the Doob-Dynkin Lemma, there exists a Borel-function $\mathbb{R}^{\mathbb{N}}:\rightarrow \mathbb{R}$ such that $Y=f(\mathbb{X})$ a.s. I am wondering if there also exists Borel-functions $f_n: \mathbb{R}^n\rightarrow\mathbb{R}$ such that
$$Y=\lim\limits_{n\rightarrow \infty}f_n(X_1,X_2,\ldots,X_n), a.s.?$$
Also, does the answer depend on wheter the random variables are independent or not?