I am working on an exercise in stochastic process stating that:
Denote $\mathbb{R}^{\mathbb{T}}$ to be the set of all functions $x:\mathbb{T}\longrightarrow\mathbb{R}$, where $\mathbb{T}$ is some index sets. Let $B\in\mathcal{B}(\mathbb{R}^{n})$, then the cylinder set is defined by $$\mathcal{C}(t_{1},\cdots, t_{n}, B):=\{x\in\mathbb{R}^{\mathbb{T}}:(x_{t_{1}},\cdots, x_{t_{n}})\in B\}.$$ Show that the set of all cylinders form an algebra.
I have some attempt, and I think I showed the closure under complement, but I was stuck in the closure under finite intersection.
Here is my attempt:
Denote $\mathfrak{C}$ to be the collection of all cylinder sets. To show the closure under complement, let $E\in\mathfrak{C}$, then $E$ can be written as $$E=\mathcal{C}(t_{1},\cdots, t_{n}, B)=\{x\in\mathbb{R}^{\mathbb{T}}:(x_{t_{1}},\cdots, x_{t_{n}})\in B\},$$ for some $B\in\mathcal{B}(\mathbb{R}^{n})$ and $t_{1},\cdots, t_{n}\in\mathbb{T}$.
Then,
\begin{align*}
E^{c}&=\{y\in\mathbb{R}^{\mathbb{T}}:(y_{t_{1}},\cdots, y_{t_{n}})\notin B\}\\
&=\{y\in\mathbb{R}^{\mathbb{T}}:(y_{t_{1}},\cdots, y_{t_{n}})\in B^{c}\},
\end{align*}
but the complement of a Borel set is still a Borel set, so the last set is still a cylinder set.
Thus, $E^{c}\in\mathfrak{C}$.
But is it necessary that if $(y_{t_{1}},\cdots, y_{t_{n}})\notin B$ then $(y_{t_{1}},\cdots, y_{t_{n}})\in B^{c}$? if so, why?
To show the closure under finite intersection, let $C_{1}, C_{2}\in\mathfrak{C}$, then $$C_{1}=\mathcal{C}(t_{1},\cdots, t_{n}, B_{1})=\{x\in\mathbb{R}^{\mathbb{T}}:(x_{t_{1}},\cdots, x_{t_{n}})\in B_{1}\},$$ $$C_{2}=\mathcal{C}(s_{1},\cdots, s_{n}, B_{2})=\{y\in\mathbb{R}^{\mathbb{T}}:(y_{s_{1}},\cdots, y_{s_{n}})\in B_{2}\},$$ for some $B_{1}, B_{2}\in\mathcal{B}(\mathbb{R}^{n})$ and $t_{1},\cdots, t_{n}, s_{1}\cdots, s_{n}\in\mathbb{T}$.
Then $$C_{1}\cap C_{2}=\{z\in\mathbb{R}^{\mathbb{T}}:(z_{t_{1}},\cdots, z_{t_{n}})\in B_{1}, (z_{s_{1}},\cdots, z_{s_{n}})\in B_{2}\},$$
but what I should do next?
Thank you so much!
Best Answer
For the first question you only have to know what $B^{c}$ means. If an element does not belong to $B$ it belongs to $B^{c}$.
For the second question you need the following fact about cylinder sets: $\{(x:x_{t_1},(x:x_{t_2},...,x_{t_n}) \in B$ can be written as $\{x:(x_{t_1},x_{t_2},...,x_{t_n},x_{t_{n+1}}) \in B_1\}$ where $B_1 =B \times \mathbb R$ and $t_{n+1}$ is any point in the index set . Using this repeatedly you observe that we can always increase the indexing set $(t_1,t_2,...,t_n)$ in any cylinder set by suitably modifying the Borel set $B$. The idea now is to write the two given cylinder sets with the same indexing set (by taking the union of the given indexing sets) and this makes it obvious that their union/intersection is also a cylinder set.