 Solution

The span of the $n$ asset payoff functions has dimension at most $n$. If there were $n+1$ disjoint measurable events $A_1,\dots ,A_{n+1}$ of positive probability, their indicator functions would be linearly independent as random variables modulo almost-sure equality. Indeed, restricting a zero linear combination to $A_j$ forces its $j$th coefficient to vanish. Market completeness would put all these independent functions inside a span of dimension at most $n$, a contradiction.

This implies the stronger meaningful partition conclusion: terminal information has at most (n) positive-probability atoms. Start with the whole sample space and split any event which is not a probability atom of a measure into two positive-probability measurable subsets. Each split increases the number of disjoint positive events, so no more than $n-1$ splits are possible. The resulting partition has $m\le n$ components, and each is a probability atom of a measure, since otherwise another split would be possible. Null sets can be included in a component without altering any random variable modulo null sets.

On such an atom every measurable real-valued random variable is constant almost surely: if its distribution on that atom were not concentrated at one value, an appropriate level set would split the atom. Hence every terminal claim is described by its $m$ values. This proves the finite branching bound in a complete market for one period, rather than the vacuous weaker observation that the whole space itself is one event.