The cylinder sigma-algebra need not be the full path-space power set

Assuming countable choice, for uncountable $I$ every ${\mathcal{C}}_I$-event is determined by countably many coordinates (Assuming countable choice, cylinder-measurable events depend on only countably many coordinates). Thus a subset that is not determined by any countable coordinate set is not in ${\mathcal{C}}_I$; such subsets occur, for example, in uncountable Bernoulli path spaces. Consequently the extension theorems in general construct measures only on ${\mathcal{C}}_I$, not on every subset of path space and not for every set-defined path functional. If every coordinate space is a singleton, by contrast, the path space is a singleton and ${\mathcal{C}}_I$ is its full power set. The companion counterexample makes the proper-inclusion boundary concrete.