To pose the question let us start by recalling the following notions:

*Transition Probabilities*. A*transition probability matrix*between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$ is a function $P:S\times\mathcal{V}\to [0,1]$ such that for every $s\in S$, $P(s,\cdot)$ is a probability measure in $\mathcal{V}$ and for every $A\in\mathcal{V}$, $P(\cdot,A)$ is $\mathcal{S}-$measurable. If $(S,\mathcal{S})=(V,\mathcal{V})$ we will speak of a transition probability matrix*in*$(S,\mathcal{S})$.*Random Elements*. A*random element*of a measurable space $(S,\mathcal{S})$ is a $\mathcal{F}/\mathcal{S}$ measurable function $\xi:\Omega\to S$ defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ (we can identify two random elements if they coincide at $\mathbb{P}-$a.e. $\omega$, but this definition will be enough for our purposes). Note that, in this definition, the word*random*comes from the introduction of a probability measure in $(\Omega,\mathcal{F})$, but this measure has no role in the property defining $\xi$.*Markov Chains (with fixed state space).*A sequence $(\xi_{k})_{k\in\mathbb{Z}}$ of random elements of $(S,\mathcal{S})$ defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$ is a*Markov chain*if for every $k\in\mathbb{Z}$ there exists a transition probability matrix $P_{k}$ in $(S,\mathcal{S})$ such that for every $A\in\mathcal{S}$, $$\omega\mapsto {P}_{k}(\xi_{k}(\omega),A)$$ defines a version of $\mathbb{P}[\xi_{k+1}\in A|\sigma(\xi_{j})_{j\leq k}]:= E[I_{A}\circ \xi_{k+1}| \sigma(\xi_{j})_{j\leq k}]$ (the conditional expectation is taken with respect to $\mathbb{P}$), where $I_{A}$ is the indicator (or "characteristic'') function of $A$. This implies in particular that for every $A\in\mathcal{S}$ and every $k\in\mathbb{Z}$ $$\mathbb{P}[\xi_{k+1}\in A|\sigma(\xi_{j})_{j\leq k}]=\mathbb{P}[\xi_{k+1}\in A|\sigma(\xi_{k})].$$*Generalized (?) Markov Chains.*If we can verify the last equation (regardless of whether the family of transitions matrices $(P_{k})_{k\in\mathbb{Z}}$ as before exists), we will say that $(\xi_{k})_{k\in\mathbb{Z}}$ is a*generalized*Markov chain.

**Question:** *Is every generalized Markov chain a Markov chain?* I.e. can we always find, for a generalized Markov chain, a family of transition probability matrices $(P_{k})_{k\in\mathbb{Z}}$ such that $\omega\mapsto P_{k}(\xi_{k}(\omega),A)$ is a version of $\mathbb{P}[\xi_{k+1}\in A|\sigma(\xi_{j})_{j\leq k}]$?

Again, I don't know if this question is elementary or its answer is well-known. If such is the case, I would appreciate a reference/explanation of the answer before it gets closed. Note also that the question can be generalized easily to the case in which the state spaces vary with $k$, but an answer to this version is sufficient for my purposes.

Thanks for your attention!