= Solution
The span of the $n$ asset payoff functions has <dimension> at most $n$. If there were $n+1$ disjoint measurable events $A_1,\ldots,A_{n+1}$ 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: \b[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.
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