Some preliminaries for the canonical construction of a Brownian Motion, help needed.

Question

I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could help me, I spent already a lot of time with recommended books, but most of the time, they contain even less information as I was given in the lecture.

So we consider the space $$C(\mathbb{R_+},\mathbb{R})=\{f:\quad f:\mathbb{R_+} \longrightarrow \mathbb{R} \text{ is continuous}\}$$ and we put the "natural" topology on this space, which is the one of uniform convergence on compact sets and we define $\mathscr{C}$ to be the Borel-$\sigma$-field associated with it. Now here are the facts which I do not really understand how to obtain:

1. A typical element of $C(\mathbb{R_+},\mathbb{R})$ writes $w=\{w(t):t\geq0\}$.
2. The Borel-$\sigma$-field $\mathscr{C}$ is generated by cylindrical sets of the type $$A=\{w\in C(\mathbb{R_+},\mathbb{R}) : \quad w(t_1)\in B_1, ...,w(t_M)\in B_m\}$$ where $(t_1,...,t_m)\in R_+^m$ and $B_1,...,B_m\in\mathscr{B}(\mathbb{R})$.
3. The class $P$ of cylindrical sets is a $\pi$-system, i.e. $C(\mathbb{R_+},\mathbb{R}) \in P$ and $P$ is closed with respect to finite intersections. (I have never seen before the definition of a $\pi$-system, I looked it up, but I do not understand why we require here that $C(\mathbb{R_+},\mathbb{R}) \in P$)

So I hope there is anybody who can clear things up for me, I would be very glad, any help is welcome!

Thanks in advance!

Answer 1

1. A "typical element" of $C(\mathbb{R}_+,\mathbb{R})$ is simply a function $w: \mathbb{R}_+ \to \mathbb{R}$. The equation $w = \{w(t); t \geq 0\}$ is a sloppy notation for this fact (the left-hand side is a function whereas the right-hand side is the image of the function).
2. Note that the topology of compact convergence is metrizable, let's call the metric $d$. One can actually show $$d(v,w) = \sum_{n=1}^{\infty} \left(1 \wedge \sup_{0 \leq t \leq n} |w(t)-v(t)| \right) \cdot 2^{-n}$$ does the job. Moreover, denote by $\mathcal{P}$ the set of all cylindrical sets. Note that $\sigma(\mathcal{P})=\sigma(\pi_t; t \geq 0)$ where $\pi_t: C(\mathbb{R}_+,\mathbb{R}) \to \mathbb{R}, w \mapsto w(t)$ denotes the projection.
   • $\sigma(\mathcal{P}) \subseteq \mathcal{S}$: Let $A=\{w \in C; \forall j=1,\ldots,n: w(t_j) \in B_j\} \in \mathcal{P}$. Then $$A = \bigcap_{j=1}^n \pi_{t_j}^{-1}(B_j)$$ We have $$|\pi_t(w)-\pi_t(v)| = |w(t)-v(t)| \leq 2^n \cdot d(v,w)$$ for all $n \geq t$. This shows that $\pi_t$ is continuous, hence measurable, thus $\pi_t^{-1}(B) \in \mathcal{S}$ for arbritary $t \geq 0$, $B \in \mathcal{B}(\mathbb{R})$. This implies $A \in \mathcal{S}$ and therefore $\sigma(\mathcal{P})\subseteq \mathcal{S}$.
   • $\mathcal{S} \subseteq \sigma(\mathcal{P})$: Let $v,w \in C(\mathbb{R}_+,\mathbb{R})$, then $$d(v,w) = \sum_{n=1}^{\infty} \left( 1 \wedge \sup_{t \in [0,n] \cap \mathbb{Q}} |\pi_t(v)-\pi_t(w)| \right) \cdot 2^{-n}$$ since $v,w$ are continuous. This shows that the mapping $v \mapsto d(v,w)$ is $\sigma(\mathcal{P})$-measurable. Hence $B(w,r) \in \sigma(\mathcal{P})$ for all $r>0$. Since $(C(\mathbb{R}_+,\mathbb{R}),d)$ is seperable (i.e. there exists a countable dense subset), we conclude $\mathcal{S} \subseteq \sigma(\mathcal{P})$.
3. It's not difficult to prove that $\mathcal{P}$ is closed under finite intersections. Let's consider two sets $A,B \in \mathcal{P}$, then $$A = \{w; \forall j=1,\ldots,m: w(s_j) \in B_j\} \qquad \qquad B=\{w; \forall j=1,\ldots,n: w(t_j) \in C_j\}$$ for Borel sets $C_j, B_j \in \mathcal{B}(\mathbb{R})$, $s_j,t_j \geq 0$. Thus $$A \cap B = \{w; \forall j=1,\ldots,n+m: w(s_j) \in B_j\}$$ where we defined $$s_j := t_{j-m} \qquad \qquad B_j := C_{j-m}$$ for $j > m$. Obviously, $A \cap B \in \mathcal{P}$. Similar proof works for finite intersections.