Supposing $X_t$ is a Markov Process, can the transition kernel be defined by $K_t(x,A):= P(X_{t+1} \in A | X_t = x)?$ Assume that $X_t : \Omega \to \mathbb{R}^n$.
The issue is that under the normal definition of conditional probability, r.h.s is defined as $P(X_{t+1} \in A | X_t = x) =\frac{P( (X_{t+1} \in A) \cap (X_t = x))}{P(X_t = x)}$ and the denominator is zero for most random variables. Even if this is assumed to be $E[I_A(X_{t+1}) | \sigma(X_t = x)]$, the conditional expectation can take arbitrary values on the set $\{X_t =x\}$ if $P(X_t = x) =0$.
Another definition I could gather from the web is that $K_t(x,A)$ is called a transition kernel if $K_t(X_t(\omega),A) := E[I_A(X_{t+1})|X_t](\omega)~\forall \omega \in \Omega.$ Also, $K_t(x,\cdot)$ should be a probability measure so that the conditional expectations should be regular (if I am not wrong).
The book (page 18) I am reading uses the first definition given above.
Thanks for the help.