Proposition 8

Let $(A_i)_{i\in I}$ be a denumerable family of pairwise disjoint universally measurable subsets of Sp($u$), and let $p_i$ be the spectral projector of $u$ defined by $A_i$. The union A of the sets $A_i$ is a universally measurable subset of Sp($u$), and the series $\sum_ip_i$ converges to $p_A$ in $\mathscr{L}(E)$ endowed with the topology of simple convergence.

The set A is universally measurable by INT, IV, p. 177, § 5, n$^o4$, cor. 2. The series $\sum_i\varphi_{A_i}$ converges simply to $\varphi_A$ and its partial sums are bounded by 1. The assertion therefore results from prop. 6 of IV, p. 276.