# Some Questions on Construction of Wiener Measure/Wiener Process

To pose my question, I first have to describe the construction which I use (due to Polletta): Assume that $$\Omega = \Pi_{t \in [0, \infty)} \dot{\mathbb{R}}$$ with topology of uniform convergence and sigma algebra $$\Sigma$$ generated by this topology. The way one can build a Wiener measure defined on the subset of continuous functions on this space is as follows: Let $$\Omega'= \Pi_{t \in \mathbb{Q}^+} \dot{\mathbb{R}}$$ and its sigma algebra $$\Sigma'$$ the sigma algebra generated by simple cylinders of the form $$C'(t_1,...,t_n,a_1,b_1,...,a_n,b_n)=\{\omega(t_i) \in [a_i,b_i)\}$$. We can define the Wiener measure on simple cylinders as $$\mu'(C'(T,A,B))= \prod_{i=1}^n \int_{a_i}^{b_i} g(x_i,t_{i+1}-t_i)dx_i$$ with g being the Gaussian distribution. This can be extended to $$\Sigma'$$. Defining Holder continuous functions on $$\Omega'$$ (denoted as $$\Omega'_c$$) one shows that this set has measure 1. Then define the embedding $$E:\Omega' \rightarrow \Omega$$ by: if continuous take it to is extension, if not take it $$0$$. Show that $$E$$ is measurable with respect to $$\Sigma'$$, $$\Sigma$$. This allows you to define a probability measure on $$\Omega$$ by $$\mu(A)=\mu'(E^{-1}(A))$$.

My first question is if I wanted to write down the measure for a cylinder set say $$C(t,U) \in \Omega$$ how would I do that? When I take $$t \in \mathbb{Q}$$, $$E^{-1}(C(t,U)) \neq C'(t,U)$$ since the inverse image does not contain the discontinuous functions, however discontinuous functions have measure $$0$$ so one has that the measure of this set should be $$\mu(C'(t,U))$$ which we know what it is. But if we take $$t$$ irrational, I would expect $$\mu(E^{-1}(C(t,U)))$$ still to be equal to $$\int_{U}g(x,t)dx$$. For this I tried writing $$E^{-1}(C(t,U))$$ as a limit of decreasing sets but could not. It is equal to $$\{ \omega' \in \Omega'_c | \lim_{t_i \rightarrow t} \omega'(t_i) \in U\}$$. Even when you attempt to write this as an intersection of sets, I could not find a way in which they would be decreasing and lead to some sort of limit. Could it could be written as some sort of "limit" $$C(t_i,U)$$ where $$t_i \rightarrow t$$ where $$t_i$$ are in $$\mathbb{Q}$$

My second question is how would you compute the expectation of a random variable that is not constant on the cylinders? Those which are constant is easy to do by writing $$\Omega$$ as union of cylinders and then using the measure define on each cylinder (like $$X(\omega)=\omega^2(t)$$)). I tried thinking about $$X(\omega)=\int_{0}^t\omega^2(s)ds$$ but could not even see if it was measurable or not.

Thanks

Okay, I think I got the answer. For $$t$$ irrational, let $$t_n$$ be an increasing sequence of rationals converging to $$t$$ and define the sets $$C'_n = C'(t_n,U)$$. If $$\gamma \in C'(t,U)$$ (defined using limits in $$\Omega'$$) then one can s,how that $$\gamma \in \text{liminf}(C'_n)$$ (assuming $$U$$ is open if $$\gamma(t) \in U$$ then for all $$t_n$$ large enough $$\gamma(t_n) \in U$$). So $$C'(t,U) \subset \text{liminf}(C'_n)$$. On the other hand of $$\gamma \in \text{limsup}(C'_n)$$ then, $$\gamma(t_n) \in U$$ for infinitely many $$t_n$$. Taking the limit $$\gamma(t) \in U$$ so $$\text{limsup}(C'_n) \subset C'(t,U)$$. This proves set theoretic limit of $$C'_n$$ is $$C(t,U)$$ hence $$p_{\Omega}(C'(t,U))=\lim_{n \rightarrow \infty} C'(t_n,U) = \int_{U} g(x,t) dx$$.
For the second one assume $$X(\gamma)=\int_0^t (\gamma(s))^2ds$$. Let $$X_n(\gamma)=\frac{1}{n}\sum_{i=1}^n \gamma(\frac{ti}{n})$$. Then $$X_n(\gamma)=f_n \circ M_{\frac{1}{n},...,t}$$ where $$M_{t}$$ is the process map of the Wiener process and $$M_T=M_{t_1,...,t_n}=(M_{t_1},...,M_{t_n})$$ and $$f(x_1,...,x_n)=\frac{1}{n}\sum_{i=1}^n x^2_i$$. So $$\mathbb{E}(X_n)=\int_{\Omega}f_n \circ M_T dp_{\Omega}= \int_{\mathbb{R}^n}(\frac{1}{n}\sum_{i=1}^n x^2_i) g(\frac{1}{n},x_1)g(\frac{1}{n},x_2-x_1)....$$ One can do the calculations and show that this is actually equal to $$\mathbb{E}(X_n)=\frac{1}{n}\sum_{i=1}^n \mathbb{E}(M^2_{\frac{ti}{n}})$$ which converges to the integral $$\int_{0}^t \mathbb{E}(M^2_s)ds=\int_0^t s = \frac{t^2}{2}.$$ I guess this could be derived using additivity of $$E$$ as well but $$X$$ was just a particular choice I wanted to see how to calculate expectations of more complex random variables and now I see that it is not really an issue whether if it is constant on cylinders or not.