Exercise 2.2.7 of Karatzas and Shreve

Question

In the book by Karatzas and Shreve there is an exercise with a beneficial implication.

I report the exercise here (and, at the end of the post, a summary of the nomenclature, just for completeness).

Execise Show that the only $\mathcal{B}\left(\mathbb{R}^{[0,\infty)}\right)$-measurable set contained in $C([0,\infty])$ is the empty set. (Hint: A typical set in $\mathcal{B}\left(\mathbb{R}^{[0,\infty)}\right)$ has the form $$ E=\{\omega\in\mathbb{R}^{[0,\infty)}|(\omega(t_1),\dots,\omega(t_n)\in A) $$ where $A\in\mathcal{B}(\mathbb{R}\times\cdots\times\mathbb{R})$).

It is clear, from the hint, that the authors are suggesting to consider sets of the form $A=A_1\times\cdots \times A_n$, with $A_j$ a Borel set of $\mathbb{R}$. My guess is that, with some proper choice, one can prove that $\omega$ cannot be continuous. But I can't really proceed, so probably this is not the smartest way.

Nomenclature

1. $\mathcal{B}(\mathbb{R}^n)$ = The Borel sigma-algebra on $\mathbb{R}^n$
2. $\mathbb{R}^{[0,\infty)} = $ The set of all real-valued functions defined on $[0,\infty)$.
3. For any $A\in\mathcal{B}(\mathbb{R}^n)$ and instants $t_j\in [0,\infty]$, $j=1,...,n$ the set
   $$ C_n(A) =\{\omega\in\mathbb{R}^{[0,\infty)}|\left(\omega(t_1),\dots,\omega(t_n)\right)\in A\}, \ $$ is called a cylinder of dimension $n$.
4. The symbol $\mathcal{B}(\mathbb{R}^{[0,\infty)})$ indicates the sigma algebra $$ \mathcal{B}(\mathbb{R}^{[0,\infty)})=\sigma\{C_n(A)| \mathcal{B}(\mathbb{R}^n),n\in\mathbb{N}\} $$ generated by all cylinder sets of all finite dimensions.

Answer 1

Hints: There is emphasis on the word typical. The sets of the form described generate the Borel $\sigma-$ algebra. Now show that every set in the Borel $\sigma-$ algebra is of the form $ E=\{\omega\in\mathbb{R}^{[0,\infty)}|(\omega(t_1),\omega(t_2),\dots)\in A)$ for some sequence $(t_n)$ and some $A \subseteq \mathbb R^{[0,\infty)}$. [To do this show that all Borel sets which are of this type form a $\sigma-$algebra containing the ''typical'' sets mentioned by the author].

Now take any $f$ in $ E=\{\omega\in\mathbb{R}^{[0,\infty)}|(\omega(t_1),\omega(t_2),\dots)\in A)$ and observe that we can make it dis-continuous by changing the value at some number $t$ not in $\{t_1,t_2,...\}$. This finishes the proof.

This is a rigorous formulation of the following intuitive reasoning: Every Borel set in this space depends only on a countable number of coordinates and continuity is not determined by a countable number of coordinates.