# An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric

$$d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min \{ |\omega_1 (t) - \omega_2 (t)| , 1\} ).$$

Let $\omega_0 \in C[0, +\infty)$, $\epsilon \in (0,1)$. We want to show that the ball $B(\omega_0, \epsilon)$ is contained in the $\sigma$-algebra generated by the collection of finite-dimensional cylinder sets of the form

$$C= \{ \omega \in C[0, + \infty) \big| ( \omega(t_1) , \ldots, \omega(t_n)) \in A \}; \quad \quad n \geq 1, \quad \quad A \in \mathcal{B} ( \mathbb{R}^n),$$ where $t_1 , \ldots, t_n \in [0, +\infty)$.

Can I argue as follows (in response to the hints provided below):

Let $n_0$ be an integer such that $\frac{1}{n_0} < \epsilon$. Also, let $[0, n] \cap \mathbb{Q} = \{ t^{(n)}_1, t^{(n)}_2, \ldots \}$. \begin{eqnarray} && B( \omega_0 , \epsilon) \\ & = & \bigcup_{ l \geq n_0} \bigcap_{k \in \mathbb{N}} \bigcap_{p \in \mathbb{N}} \bigg\{ \omega \bigg| \sum_{n=1}^k \frac{1}{2^n} \max_{t \in \{t^{(n)}_1, \ldots t^{(n)}_p \}} \bigg[ |w(t) - \omega_0 (t)| \wedge 1 \bigg] \leq \epsilon - \frac{1}{l} \bigg\} \end{eqnarray} If we set $f: \mathbb{R}^p \rightarrow \mathbb{R}$ by $$f(x_1 , x_2 , \ldots , x_p) = \sum_{n=1}^k \frac{1}{2^n} \max_{i \in \{1, 2, \ldots ,p \}} \bigg[ | x_i - \omega_0 (t^{(n)}_i)| \wedge 1 \bigg],$$ which is Borel-measurable. Then, $$B( \omega_0 , \epsilon) = \bigcup_{ l \geq n_0} \bigcap_{k \in \mathbb{N}} \bigcap_{p \in \mathbb{N}} \bigg\{ \omega \bigg| \big( \omega (t^{(n)}_1) , \omega(t^{(n)}_2), \ldots, \omega(t^{(n)}_p) \big) \in f^{-1} \bigg( \bigg( -\infty, \epsilon - \frac{1}{l} \bigg] \bigg) \bigg\}$$

• Hints: (i) you get the same metric if you write $\sup_{t\in[0,n]\cap\Bbb Q}$ in place of $\max_{t\in[0,n]}$. (ii) $a<b$ if and only if there exists $N$ with $a\le b-1/N$. (iii) If $a_n$ is increasing then $\lim a_n\le b$ if and only if $a_n\le b$ for all $n$. Commented Dec 27, 2015 at 17:11
• @DavidC.Ullrich Is my above reasoning correct? Commented Dec 28, 2015 at 5:41
• Don't have time to read it carefully. The result seems like it should be true, and a proof should look something like what you wrote... Commented Dec 28, 2015 at 14:38