Projection-valued measure

Remarks

Clause 4 is equivalent to the strong countable additivity: if $B = \bigcup_n B_n$ with $B_n \in \Sigma$ pairwise disjoint, then $E(B) x = \sum_n E(B_n) x$ for every $x \in H$. By clause 3, the projections of disjoint sets have orthogonal images, and all $E(B)$ commute. For each $x$, $E_{x,x}$ is a positive measure of total mass $\norm{x}^2$. Example: let $H = \ell^2$ with orthonormal basis $(e_n)_{n \ge 1}$, let $\Omega = \R$ with its Borel sets, and let $(\lambda_n)$ be real numbers. Let $E(B)$ be the orthogonal projection onto the closed span of $\set{e_n : \lambda_n \in B}$; clauses 1 to 4 follow from the fact that $E_{x,y}(B) = \sum_{n : \lambda_n \in B} x_n \overline{y_n}$. Non-example: on $\Omega = \R$, let $E(B) = P$ if $0 \in B$ and $E(B) = 0$ otherwise, for an orthogonal projection $P \ne I$; then $E(\R) = P \ne I$, so clause 2 fails.