Proof of (Strong) Markov Property using sigma-algebras

Question

I would like to ask if any of you know of a good resource containing rigorous proof (using sigma-algebras) of Markov Property and Strong Markov Property respectively in terms of Discrete Time Markov Chains.

(All resources I have found dealt with Continuous Time Markov Chains or with discrete MP/SMP in terms of Brownian Motion.)

Thanks, C.

Answer 1

Varadhan's "Probability Theory" has a proof of the SMP for discrete-time chains.

Answer 2

See Theorem 6.3.4. of Probability: Theory and Applications (4th edition) by Richard Durrett.

Also, a very accessible proof is found in Theorem 1.4.2 of Markov Chains by J.R. Norris.