= Sigma-algebra of a finite-valued random vector
{title2=$|\sigma(V)|=2^{|V(\Omega)|}$}
Let a <random vector> $V$ have a finite actual range of $q$ values. Its nonempty fibres partition the <sample space>, and every <event> in $\sigma(V)$ is a <union> of those fibres. Thus each fibre is an <atom of a sigma-algebra> and $\sigma(V)$ has exactly $2^q$ elements. This assertion concerns the actual range; additional values attained only on <null sets> must still be counted when working with an uncompleted <sigma-algebra> literally.
To verify the claim, the fibres belong to $\sigma(V)$ as preimages of singleton <Borel sets>. Their <unions> form a finite <sigma-algebra> making $V$ <measurable>, so minimality of the generated <sigma-algebra> gives the reverse inclusion. Distinct subcollections have distinct <unions> because the fibres are nonempty and disjoint.
Consequently a finite history of finite-valued observations generates a finite <natural filtration> at each fixed time. It may be strictly smaller than the <filtration> of its driving continuous observations: a single <uniform random variable> already generates infinitely many distinct <events> $\{U\leq t\}$ with different <probabilities>.
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