In office hours, a professor mentioned that an injective transformation of a Markov process remains Markov. Intuitively, this makes sense to me as you can "recover" the original Markov process from the new one, but I don't understand how to formalize this. How can this statement be proved?
2026-03-27 23:14:00.1774653240
Injections of Markov Processes
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$$\bar\mu_0(u)=\mu_0(i^{-1}(u))\qquad\bar P(u,v)=P(i^{-1}(u),i^{-1}(v))$$