I study probability theory, and I'm stuck on the problem below.

$T$ : compact pseudo-metric space ($d(x,y)=0 \nRightarrow x=y$)
$S$ : dense subset of $T$
$C(T)$ : a set of all continuous functions on $T$
A compact pseudo-metric space is separable.

I know the Borel $\sigma$-algebra on $C(T)$ is the smallest $\sigma$-algebra making all projections $z\mapsto z(s)$ measurable ($s\in S,z\in C(T)$), but I would like to ask you how to prove it.

There is a hint.
First, prove the closed ball around $z_0$ of radius $r $ equals $\bigcup_{s\in S_0}\{z:|z(s)-z(s_0)|\leq r\}$ , where $S_0$ is a countable dense subset of $S$ .
Then, prove every closed and open ball is projection measurable .
Finally, prove the statement of this proposition.

  • $\begingroup$ Where exactly did you get stuck? Did you justify the first part of the hint? $\endgroup$ – Kavi Rama Murthy Sep 16 at 12:49
  • $\begingroup$ I tried, but did not justify any part of them. $\endgroup$ – Joey Sep 16 at 12:54
  • $\begingroup$ @KaviRamaMurthy So first part . $\endgroup$ – Joey Sep 17 at 7:50

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