I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated by the cylindrical sets: $$Cly:=\{x\in C([0,\infty),\Bbb{R}^d)|x(t_1)\in A_1,...,x(t_n)\in A_n\}$$ Here $A_i$ are Borel sets of $\Bbb{R}^d$. I also know the action of $\mu$ on $A\in Cly$ is given by: $$\mu[A] =\prod^n_{i=1} (2\pi(t_i-t_{i-1}))^{-d/2} \int_{A_1}...\int_{A_n} \exp\left( -\frac{1}{2} \sum^n_{i=1} \frac{|x_i-x_{i-1}|^2}{t_i-t_{i-1}}\right) dx_n...dx_1$$ My question is how do we go from this to the integral of a function $f(\omega)$ to (as indicated in (1)): $$\lim_{n\rightarrow \infty}\left(\frac{n}{4D\pi t}\right)^{nd/2}\int_{\Bbb{R}^d}...\int_{\Bbb{R}^d}\hat f(x_1,...,x_n)\exp\left( - \sum^n_{i=1} \frac{|x_i-x_{i-1}|^2}{4Dt/n}\right) dx_n...dx_1$$ (1) also explains that the existence of the measure is to do with the Riesz Representation Theorem - how is this the case?

**Sources**

(1) Nathanson, E.S., 2014. *Path integration with non-positive distributions and applications to the Schrödinger equation*. The University of Iowa. pdf pg$\sim$27 actual pg$\sim$17