model more complicated sets of quantum mechanical observables.

Let ${\ensuremath{\mathcal S}}$ be a set of Hermitian operators on a finite-dimensional Hilbert space , and let $(\Omega,{\ensuremath{\mathrm{CO}}})$ be an observation frame. By a *model* of ${\ensuremath{\mathcal S}}$ in $(\Omega,{\ensuremath{\mathrm{CO}}})$ we mean a partial function $\mu$ from closed subspaces of  to subsets of $\Omega$ such that:

1.  For each operator $A\in{\ensuremath{\mathcal S}}$ and each sum $E$ of eigenspaces of $A$, $\mu(E)$ is defined and is an observable event in $(\Omega,\mathrm{CO})$.

2.  For each operator $A\in{\ensuremath{\mathcal S}}$, the collection $\mathcal E_A$ of all sums of eigenspaces of $A$ has $\{\mu(E):E\in\mathcal E_A\}$ coobservable in $(\Omega,\mathrm{CO})$.

3.  If $E$ is a sum of eigenspaces of some $A\in{\ensuremath{\mathcal S}}$, then the complementary sum $E^\bot$ satisfies $\mu(E^\bot)=\Omega-\mu(E)$.

4.  If $E_1$ and $E_2$ are sums of eigenspaces of a single $A\in{\ensuremath{\mathcal S}}$, then $\mu(E_1+E_2)=\mu(E_1)\cup \mu(E_2)$.

> [[Q: ]{}]{}Explain M1. Why are we talking about sums of eigenspaces?
>
> [[A: ]{}]{}Given a set $V$ of eigenvalues of a Hermitian operator $A\in{\ensuremath{\mathcal S}}$, let $E$ be the sum of the corresponding eigenspaces of $A$. As a closed subspace of the Hilbert space, $E$ is an event. When we measure the Hermitian operator $A$, the result $v$ is one of 