Is the set of continuous functions measurable?

Question

Let $X$ be a separable metric space.

Then, is $C([0,\infty),X)\in \otimes_{t\in [0,\infty)} \mathscr{B}_X$?

I want to write what I have tried at least, but I have no idea how to approach this kind of problem.. Thank you in advance.

Answer 1

The answer is no. Consider for example $X=\mathbb{R}$. The idea is that the cylinder generated $\sigma$-field is not "big enough" to determine whether a function is continuous or not. Specifically, you can prove the following:

Let $A\in\mathscr{B}\big(\mathbb{R}^{[0,\infty)}\big)$. There exists a set $T\subset[0,\infty)$ which is at most countable and such that for all $\omega\in\mathbb{R}^{[0,\infty)}$ and $x\in A$ the following holds: $$\forall\;t\in T: \omega(t)=x(t)\;\;\Rightarrow \omega\in A $$

i.e. you can basically determine whether a function lies in a measurable set by checking if it coincides with a function from the desired set in countably many places. From this, you can deduce by contradiction that the set of continuous real-valued functions can not be measurable.