This is the exercise 8.14 of Donald L. Cohn's measure theory. There he provided a counterexample to show that the pointwise limit of measurable function may not be measurable if the range is not metrizable.
Let $(X,\mathcal{A})$ be $([0,1],\mathcal{B}([0,1]))$ and let $Y$ be $[0,1]^{[0,1]}$ with the product topology. For each $n$ let $f_n: X\to Y$ be the function that takes $x$ to the element of $Y$ (i.e., to the function from $[0,1]$ to $[0,1]$) given by $t\mapsto \max (0,1-n|t-x|)$.
If I'm not going wrong, $f_n$ converges to $t\mapsto \mathbb{1}_{x=t}(x)$. However, this limit function seems measurable, which is not a counterexample. where do I go wrong?
Let $B$ be any non-Borel set in $[0,1]$. Consider $\{\phi \in [0,1]^{[0,1]}:\phi (t)>1/2$ for some $t \in B\}$. This set is an open set in $[0,1]^{[0,1]}$. The inverse image of this set under $f$ is precisely $B$, Hence $f$ is not measurable.