Exercise 1, § 5

Let E be a Hilbert space and $\mathscr{P}(E)\subset \mathscr{L}(E)$ the set of orthogonal projections in E. Let S be a universally measurable subset of $\mathbf{C}$. We denote by $\mathscr{T}(S)$ the set of universally measurable subsets T of $\mathbf{C}$ such that $T\subset S$. A measure on S with values in the projections of E is a mapping $\varpi$ of $\mathscr{T}(S)$ into $\mathscr{P}(E)$ such that

(i) We have $\varpi (\emptyset ) = 0$ and $\varpi (S) = 1_E$.

(ii) If $(T_n)$ is a sequence of pairwise disjoint sets in $\mathscr{T}$ (S), and if T is the union of the sets $T_n$, then

in the space $\mathscr{L}(E)$ endowed with the topology of simple convergence.

For every normal partial operator $u$ on E with spectrum equal to S, and for $T\in \mathscr{T}$ (S), we denote by $p_{T,u}$ the spectral projector of $u$ defined by T.

The mapping which to a normal partial operator $u$ on E of spectrum S associates the mapping $T\mapsto p_{T,u}$ is a bijection between the set of normal partial operators of spectrum S and the set of measures on S with values in the projections of E.