Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 3 / b / Solution

Every indicator random variable must lie in the linear span of the $n$ terminal asset prices. If the sigma-algebra had $n+1$ disjoint events of positive probability, their indicators would be linearly independent, yet completeness would place all of them in an at-most-$n$-dimensional space. Hence, modulo null events, the sample space has at most $n$ positive-probability atoms.