Let $T=[a,b]$, $a<t_0<b$ Prove that the following sets do not belong to $B(R^T)$
$$A_1=(x\in{R^T}:sup_{t\in[a,b]}|x_t|\leq1)$$
$$A_2=(x\in{R^T}:t\to x_t\space continous\space on\space [a,b])$$
$$A_3=(x\in{R^T}:lim_{t\to t_0}x_t=0)$$
Here comes the part I dont understand totally, prove that if trajectory ($t\to X_t$) is continous then those sets belong to $C(T)\cap B(R^T)$, the definitions are as follows: $B(R^T)=\sigma(\{x\in R^T:x(t)\in A\},A\in B(R),t\in T)$ $\\$
$C(T)$ is a space of continous functions on $T$ with supremum norm.
The only solution that pop ups to my mind is that the set [a,b] is uncountable and there is a theorem which says that $B(R^T)$ may be expressed as sum over $t_1,t_2,...$ any clarification on this would be very nice, thanks
Hint: show that every set in $B(\mathbb R^{T})$ is of the type $\{x:(x_{t_1},x_{t_1},...) \in E\}$ for some Borel set $E$ in $\mathbb R^{\mathbb N}$ and some sequence $\{t_1,t_2,...\}$. [For this consider the collection of all sets in $B(\mathbb R^{T})$ which are of this type and prove that this is a sigma algebra]. The non-measurabilty of the given sets is easy to prove using this.