Product Measure
Definition
Given two measures $\mu$ on $(S_1, \mathcal{F}_1)$ and $\nu$ on $(S_2, \mathcal{F}_2)$, the product measure $\mu \otimes \nu$ on $(S_1 \times S_2, \mathcal{F}_1 \otimes \mathcal{F}_2)$ satisfies
$$ (\mu \otimes \nu)(A \times B) = \mu(A)\, \nu(B) $$for all measurable sets $A \in \mathcal{F}_1$, $B \in \mathcal{F}_2$.
Role in Markov Chains
For two independent Markov chains with stationary distributions $\pi$ and $\pi'$, the joint chain has stationary distribution $\pi \otimes \pi'$.