# 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 $$(K, \mathcal{A})$$ with transition probability measures given by $$P_u$$ for each $$u \in K.$$

A distribution $$Q$$ on $$(K, \mathcal{A})$$ is called stationary if one step from it gives the same distribution, i.e., for any $$A \in \mathcal{A},$$
$\int_A P_u(A) \, dQ(u) = Q(A).$

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

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

A distribution $$Q$$ on $$(K, \mathcal{A})$$ is called stationary if one step from it gives the same distribution, i.e., for any $$A \in \mathcal{A},$$
$\int_K P_u(A) \, dQ(u) = Q(A).$

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.

• Sorry, outside my field.