a quick notational doubt clearing in the definition of Markov chain on uncountable state space

Question

1. Equation 7.18 or next few I found the notation $\int K(x_k, dx)$, my question is it same as $\int K(x_k, A)dx$ or the $\int K(x_k, \cdot)dx$ is much better? Thank you for any explanation or how to get understand that notation because I always saw from school days or later stages: $\int\text{something } dx$ rather than $\int(\text{something }, dx)$

2. Also, if one can show some intuition behind the notation of the definition of $n$ step transition kernel, that would be great, I am bit confused.