Let $X_{0},X_{1},\dots$ be a sequence of Borel spaces and, for $n=0,1,\dots$ define $Y_{n}:=X_{0}\times\dots \times X_{n}$ and $Y:=\prod_{n=0}^{\infty}X_{n}$. Let $\nu$ be an arbitrary probability measure on $X_{0}$ and for every $n=0,1,\dots,$ let $P_{n}(dx_{n+1}\mid y_{n})$ be a Stochastic Kernel on $X_{n+1}$ given $Y_{n}$. Then there exists a unique probability measure $P_{\nu}$ on $Y$ s.t., for every measurable rectangle $B_{0}\times\dots \times B_{n}$ in $Y_{n}$

$$\begin{align*} P_{\nu}(B_{0}\times\dots \times B_{n})=&\int\limits _{B_{0}} \, \nu(dx_{0}) \int\limits _{B_{1}} \, P_{0}(dx_{1}\mid x_{0}) \int\limits _{B_{2}} \, P_{1}(dx_{2}\mid x_{0},x_{1})\\ & \dots \int\limits _{B_{n}} \, P_{n-1}(dx_{n}\mid x_{0},\dots,x_{n-1}) \end{align*}$$

^statement

Moreover, for any nonnegative measurable function $u$ on $Y$, the function $$x\mapsto \int\limits u(y) \, P_{x}(dy)$$ is measurable on $X_{0}$, where $P_{x}$ stands for $P_{\nu}$ when $\nu$ is the probability Concentrated at $x \in X_{0}$.