Puzzled by the definition of stationarity

I’m reading Santosh Vempala’s survey “Geometric Random Walks: A Survey,” and already I’m puzzled at one of the very first definitions he gives.

Define a Markov chain as a state space sigma algebra pair (K,A) with transition probability measures given by Pu for each uK.

A distribution Q on (K,A) is called stationary if one step from it gives the same distribution, i.e., for any AA,
APu(A)dQ(u)=Q(A).

This definition makes sense in words, but mathematically it doesn’t seem sound: unless Pu(A)=1 a.e. (with respect to uA), this equality can’t hold. I must be missing something…

Update:
The definition should involve integration over the whole space:

A distribution Q on (K,A) is called stationary if one step from it gives the same distribution, i.e., for any AA,
KPu(A)dQ(u)=Q(A).

That this is so can be seen from the discrete case. Just goes to show you that you have to be careful even when reading peer-reviewed articles.