"Theorem. Suppose $H$ is a separable Hilbert space. A measure on $H$ is a function $f$ that assigns a nonnegative real number to each closed subspace of H in such a way that, if $\textstyle \{A_{i}\}$ is a countable collection of mutually orthogonal subspaces of $H$, and the closed linear span of this collection is $B$, then $\textstyle f(B)=\sum _{i}f(A_{i})$.
If the Hilbert space H has dimension at least three, then every measure f can be written in the form $\textstyle f(A)=\mathrm{Tr} (WP_{A})$, where $W$ is a positive semidefinite trace class operator and $P_{A}$ is the orthogonal proj…