A question about Chapman-Kolmogorov equation I'm reading ''Functional Analysis'' - K. Yosida and at page 379 there is the following claim
"The hypothesis that the particle has no memory of the past implies that the transition probability P satisfies the equation
$$P(t, x; s, E) = \int\limits_S P(t, x; u, dy) P(u, y; s, E), \;\; t < u < s,$$
where $S$ is the entire space of the chaotic movement of the particle."
What does it mean $P(t, x; u, dy)$? Is it a notation?
Thank you!

For fixed $t,x$ and $u$, $P(t, x;u, \cdot)$ is a measure. The notation $P(t, x; u, dy)$ means: "integrate with respect to that measure using $y$ as dummy variable".
