A certain formulation of the Chapman-Kolmogorov equation. I am reading a book by Taira called Semigroups, Boundary Value Problems and Markov Processes. It is a nice read, but there is one thing I don't understand regarding the Chapman-Kolmogorov equation. A markov transition function $p_t(x,E)$ where $E$ is in a Borel set is defined as a non-negative measure in the first variable and a Borel measure in the second. The Chapman-Kolmogorov equation is written as $$p_{t+s}(x,E)=\int_{K} p_t(x,\mathrm{d}y) p_s(y,E)$$ where E is a Borel set and x is in K. I don't understand exactly what this way of writing the integral means, and what the set $dy$ here is. Is there any other equivalent and more standard way to write the integral? In some sense, is this a composition of measures? I doubt it, but do you think this is a misprint? This is on p.51 in the book. Thank you!

No, it is not a misprint. By definition,

$$\mu(E) := p_t(x,E)$$

defines a measure for each fixed $x$ and $t \geq 0$. The integral

$$\int_K p_t(x,dy) p_s(y,E)$$

is the integral of the mapping $y \mapsto p_s(y,E)$ with respect to the measure $\mu$, i.e.

$$\int_K p_t(x,dy) p_s(y,E) = \int_K p_s(y,E) \mu(dy).$$