Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have unique stationary distributions $\pi, \lambda$. Is it then true that the Markov chain $(X_i, Y_i)_i$ is also $\psi$-irreducible and aperiodic?

I have read that this is true for chains on countable state spaces, but for the general state space setting I run into some (product-)measure-theoretic problems. Any advice or perhaps a reference for this topic?

I am interested in the general case described above, but if it matters, I am especially interested in the case with $\mathcal{X}=\mathcal{Y}=[0,1]^m$ and $P=Q$.