For any integer $n$, any choice of $0 < t_1 < \cdots < t_n \leq 1$, and any Lebesgue measurable set, $E \in \mathbb{R}^n$ define the “cylinder” $$I = I(n;t_1;\cdots;t_n;E) := \{ \beta(·) \in C_0[0,1];(\beta(t_1),\cdots,\beta(t_n)) \in E\}.$$ Let $\mathcal{A}$ be the class of intervals containing all the $I$ for all $n,t_1,\cdots,t_n$ and all Lebesgue measurable sets $E \in \mathbb{R}^n$, then $\mathcal{A}$ is an algebra of sets in $C_0 [0, 1].$
The $I$’s are the cylinder sets upon which we will define Wiener measure, and then standard measure theoretic ideas to extend to all measurable subsets of the infinite dimensional space, $C_0 [0, 1]$. In fact,we define its measure as $$ \mu(I) = \dfrac{1}{ \sqrt{(2\pi)^n t_1(t_2 - t_1) \cdots (t_n - t_{n-1})}} \int \cdots \int_E e^{\frac{-u_1^2}{2t_1} - \frac{(u_2 - u_1)^2}{2(t_2 - t_1)} - \cdots - \frac{(u_n - u_{n-1})^2}{2(t_n - t_{n-1})}}du_1 \cdots du_n. $$
The extension of this measure, creates a probability measure called Wiener measure. I'm trying to understand what the conditional Wiener measure is and how it works. I'm trying to get some relationship between the Wiener measure of the following sets
$$ A = \{ x(r) \in E, 0 \leq r \leq t | \, x(0) = x \, , x(t) =y \}$$ $$ B =\{ y(l) \in E, 0 \leq l \leq s | \, y(0) = y \, , y(s) =z \}$$ and $$ C = \{ x(l) \in E, 0 \leq l \leq t+s | \, x(0) = x \, , x(t+s) =z \} $$
I hope to get some kind of relationship $ \mu (A) \mu(B) \leq \mu (C)$. How can I perform the Wiener measure of these sets, since the cylinders are set to $t>0?$ Is it possible to do something in this direction using the sets $A$, $B$ and $C$ are cylinder intersections? I appreciate any suggestions.