An exercise in a book on unbounded selfadjoint operators reads as follows:
Let $\mathfrak A$ be a $\sigma$-algebra on $\Omega$ and $E$ be a mapping of $\mathfrak A$ into the projections of a Hilbert space $H$ such that $E(\Omega)=\text{Id}$. Show that $E$ is a spectral measure if and only if the following is satisfied:
$E(\bigcup_{n=1}^\infty M_n) = s$-$\lim_{n\to\infty}E(M_n)$ for all sequences $(M_n)\subset\mathfrak A$ with $M_n\subseteq M_{n+1}$ for all $n$. (where $s$-$\lim$ denotes the strong limit in $H$).
My question: This statement is wrong, isn't it? My counterexample: Take $E(M):=\text{Id}$ for all $M\in\mathfrak A$. Then $E(M)$ is a projection for each $M$ and $E(\Omega)=\text{Id}$ and the limit condition above is clearly satisfied. But $E$ is not a spectral measure, since it is not additive.
Is my reasoning correct? Does anybody know what this exercise should actually be?