Transforming an inhomogeneous Markov chain to a homogeneous one I fail to understand Cinlar's transformation of an inhomogeneous Markov chain to a homogeneous one. It appears to me that $\hat{P}$ is not fully specified. Generally speaking, given a $\sigma$-algebra $\mathcal A$, a measure can be specified either explicitly over the entire $\sigma$-algebra, or implicitly by specifying it over a generating ring and appealing to Caratheodory's extension theorem. However, Cinlar specifies $\hat{P}$ over a proper subset of $\hat{\mathcal{E}}$ that is not a ring.

We give the condition that $\widehat P$ is a Markow kernel, and we have that 
$$\widehat P((n,x),\{n+1\}\times E)=P_{n+1}(x,E)=1,$$
hence the measure $\widehat P((n,x),\cdot)$ is concentrated on $\{n+1\}\times E\}$? Therefore, we have $\widehat P((n,x),I\times A)=0$ for any $A\subset E$ and $I\subset \Bbb N$ which doesn't contain the integer $n+1$.
