 Here is one way in which we can manufacture a probability measure on \(L(\mathbf{H})\). Let \(u\) be a unit vector of \(\mathbf{H}\), and set \(\mu_u (P) = \langle Pu,u\rangle\). This gives the usual quantum-mechanical recipe for the probability that \(P\) will have value 1 in the state \(u\). Note that we can also express \(\mu_u\) as \(\mu_u(P) = Tr(P P_u)\), where \(P_u\) is the one-dimensional projection associated with the unit vector \(u\), i.e., \(P_u(x) = \langle x, u \rangle u\) for all \(x \in \mathbf{H}\). 