Cylinder sigma-algebra ($\mathcal{C}$)

On a product space ${\mathbb{R}}^T$, the cylinder sigma-algebra is the smallest sigma-algebra making every coordinate projection $x\mapsto x(t)$ measurable. Its generators ask finitely many coordinate questions. Every event in it depends on at most countably many coordinates: the collection of countably determined sets contains these generators and is closed under complements and countable unions. If $T$ is uncountable, the singleton consisting of the identically zero function is not such an event. Any countable coordinate test leaves an untested coordinate where a different function can be nonzero. Thus coordinate measurability alone need not capture every pathwise question.