I asked this question, the definition means that the chain will visit any open set with positive probability, now my question is what does this ''any open set'' means? Does it mean the only open sets which are in the Borel sigma-algebra? I mean suppose the state space of the chain is $\mathbb R$, and consider a Borel sigma-algebra on $\mathbb R$ by which I mean all events or set of all events related to the chain. Suppose in the Borel sigma-algebra an open set $V$ of $\mathbb R$ is not present, so the chain will never visit $V$ right?
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0 The only definition of the Borel sigma algebra I have ever seen: the Borel sigma algebra is the sigma algebra generated by open sets, i.e. the smallest sigma algebra containing all open sets.
