I cannot get the easy description of all spectral measures in the following simple exercise.
Let $H = \mathbb{C}$ and there is a $\sigma$-algebra $\mathcal{A}$ on some set $X$ such that $\mathcal{A}$ contains all single points of $X$.
I want to describe all spectral measures: $E : \mathcal{A} \rightarrow H$, where $PR$ denotes the space of orthoprojectors on $H$. Obviously, for $H = \mathbb{C}$ projector $E(\cdot)$ is either $0$ or $1$.
1) One case is pretty simple. Let there will be a point $x_0$ such that $E(x_0) = 1$, then it is easy to prove that $E(A) = \{1, \text{ if } x_0\in A, 0 \text{ if }x_0\neq A\}$ (by additivity and multiplicity).
2) If $E(x) = 0$ for all single points $x\in X$, then I don't have a good description. Maybe it is already sufficient description in such assumptions, however, I'm not sure.
If $E(\{x\})=0$ for all $x\in X$, then it follows that $E(A)=0$ for all countable subsets of $\mathcal A$. Hence, $X$ cannot be countable, as $E(X)=1$ is required. By building complements, $E(A^c)=1$ for countable $A$.
If $X$ is uncountable, then the following situation is possible: $$ \mathcal A=\{ A\subset X: \ A \text{ or } A^c \text{ is countable }\}. $$ Then $E(A)=1$ for uncountable $A$ and $E(A)=0$ for all countable $A$. Note that this $\mathcal A$ is the $\sigma$-Algebra generated by the singletons, see this question.