Orthogonal Additivity

Before invoking Gleason’s theorem, one further condition must be established: orthogonal additivity. This is the property that makes μ a measure on the projection lattice, not merely a function on individual projectors.

Lemma 1 (Orthogonal Additivity). Let {P_1, …, P_k} be mutually orthogonal projectors and let P = P_1 + … + P_k be the projector onto their joint subspace (the coarse-grained outcome “one of outcomes 1 through k occurred”).

Then μ(P) = μ(P_1) + … + μ(P_k).

Proof. The coarse-grained outcome P corresponds to not distinguishing among the mutually exclusive fine-grained outcomes P_1, …, P_k. By Axiom S, each fine-grained outcome is a distinct possibility. By Axiom B, one break occurs.

By Axiom R, the break produces exactly one outcome. The fine-grained outcomes are mutually exclusive (orthogonal projectors) and the coarse-grained outcome P is their disjunction.

The probability of the disjunction of mutually exclusive events is the sum of their individual probabilities — this is the defining property of a probability measure, which follows from Axiom B (one break = one outcome from the set of possibilities) and Axiom R (the record is definite = exactly one outcome is actualised).

Therefore μ(P) = ∑ μ(P_i). ∎

Note: Lemma 1 is what makes μ a frame function in the sense required by Gleason’s theorem.

A frame function is not merely a function on projectors that sums to 1 over resolutions of the identity — it respects the lattice structure of projections via orthogonal additivity.

Propositions 1 and 2 establish that μ is well-defined on projectors and independent of context. Lemma 1 establishes that μ is additive over orthogonal projectors. Together, these conditions are exactly the hypotheses of Gleason’s theorem.