Cylindrical Sets and Minimal Sigma Algebras I am trying to prove the following:
Let $\Omega_{\mathbb{R}}$ be the set of all functions$\omega : \mathbb{R} \to \mathbb{R}$ and $B_{\mathbb{R}}$ be the minimal
$\sigma$-algebra containing all the cylindrical subsets of $\Omega_{\mathbb{R}}$. 
Let $\Omega_{\mathbb{Z}+}$  be the set of
all functions from $\mathbb{Z}+ \to \mathbb{R}$, and $B_{\mathbb{Z}+}$ be the minimal $\sigma$-algebra containing all
the cylindrical subsets of $\Omega_{\mathbb{Z}+}$.
Show that a set $S ⊆ \Omega_{\mathbb{R}}$ belongs to $B_{\mathbb{R}}$ if and only if one can find a set $B \in B_{\mathbb{Z}+}$ and an infinite sequence of real numbers $t_1,t_2,...$ such that $S = ( \omega : (\omega(t_1), \omega(t_2),...) \in B )$.
Furthermore prove that the sets $(\omega \in \Omega  : |\omega(t)| < C$ for all t$ \in \mathbb{R})$ and $(\omega \in \Omega : \omega$ is continuous for all  $ t \in \mathbb{R})$
do not belong to $B.$ 
I'm not sure how to approach this problem...
 A: Hints: Set $$\Sigma := \big\{S \subseteq \Omega_{\mathbb{R}}; \exists B \in B_{\mathbb{Z}_+}, (t_j)_{j \in \mathbb{N}} \subseteq \mathbb{R}: S = \{\omega; (\omega(t_j))_{j \in \mathbb{N}} \in B\} \big\}.$$


*

*Verify that $\Sigma$ is a $\sigma$-algebra.

*Show that any cylindrical set is contained in $\Sigma$. Conclude that $B_{\mathbb{R}} \subseteq \Sigma$.

*For fixed $S \in \Sigma$ show that $$S = \bigcap_{k \in \mathbb{N}} \{\omega; \forall j=1,\ldots,k: w(t_j) = b_j\}$$ for suitable $b_j \in \mathbb{R}$. Deduce that $S \in B_{\mathbb{R}}$.

*Combining step 2 and 3 shows $\Sigma = B_{\mathbb{R}}$.

*Set $S:=\{\omega; \forall t \in \mathbb{R}: |\omega(t)|<C\}$ for some fixed constant $C>0$. Show that for any $B \in \mathcal{B}_{\mathbb{Z}_+}$ and any sequence $(t_j)_j$ we have $$S \subsetneq \{\omega; (\omega(t_1),\ldots) \in B\}.$$ To prove this strict inclusion, find an unbounded function $\omega \in \Omega_{\mathbb{R}}$ which satisfies $(\omega(t_1),\ldots) \in B$. Conclude that $S \notin B_{\mathbb{R}}$.

*Use a similar reasoning as in step 5 to conclude that the set of continuous functions is not in $B_{\mathbb{R}}$; you have to find a discontinuous function $\omega$ which satisfies $(\omega(t_1),\ldots \in  B)$ for given $B$ and $(t_j)_j$.

