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I found the following definition of stochastic process: A stochastic process on a countable state space V is a sequence of random variables $(S_n)_{n \in \mathbb{N}}$ taking values in V and defined on a probability space $(\Omega, \mathcal{F}, P)$.

Why is in this definition the $\sigma$-algebra associated to $V$ omitted? In order $S_n$ to be a random variable, then shouldn't $V$ be a measurable space? What would be the $\sigma$-algebra then?

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For a countable set $V$, the typical way to make it into a measurable space is to equip it with the discrete $\sigma$-algebra, i.e. its power set $2^V$, so that every subset of $V$ is measurable.

For most purposes, this is the only sensible thing to do. Usually one wants singleton sets to be measurable. In a countable space, every subset is a countable union of singletons, so this forces every subset to be measurable.

So most people would consider this the "default" $\sigma$-algebra for a countable set; if no $\sigma$-algebra is explicitly stated, then you are almost certainly meant to assume $2^V$. Similarly, if someone talks about $\mathbb{R}$ or $\mathbb{R}^n$ in a context where a measurable space is wanted, you should normally assume that they mean to use the Borel $\sigma$-algebra.

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