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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.

We prove the stronger finite branching bound in a complete market: modulo null sets, ${\mathcal{F}}_t$ is generated by at most $n^t$ atoms. Here an atom is a positive-probability event which cannot be split into two positive-probability measurable pieces.

At time zero there is one atom because ${\mathcal{F}}_0$ is trivial. Suppose ${\mathcal{F}}_{t-1}$ has at most $n^{t-1}$ atoms, and fix one such atom $A$. Any predictable process holdings over $(t-1,t]$ are constant on $A$, so the restriction to $A$ of every time-$t$ replicable payoff lies ina vector space of dimension of a vector space at most $n$. If $A$ contained $n+1$ disjoint positive-probability ${\mathcal{F}}_t$ events, their indicators would have linear independence on $A$. Market completeness would replicate each indicator, contradicting that dimension bound.

For clarity, the absence of $n+1$ disjoint positive pieces implies that $A$ is a union of at most $n$ atoms: start with $A$ and repeatedly split any non-atom into two positive pieces. Each split increases the count by one. The process must stop before the count exceeds $n$, and at termination every piece is an atom. Thus each time-$(t-1)$ atom has at most $n$ successors. Induction gives at most $n^t$ atoms at time $t$.

Each disjoint positive-probability ${\mathcal{F}}_t$ event contains at least one distinct atom. ThereforeThe argument includes $t=0$ and uses the ability to replicate the indicator claims at every intermediate maturity.