Let $(A_n)_{n \in \mathbb{N}}$ be a sequence of commuting self-adjoint operators on a Hilbert space, which is uniformly bounded in operator norm. We assume that the sequence converges, in the strong operator topology, to a self-adjoint, bounded operator $A$.
Let $(E_n)_{n \in \mathbb{N}}$ be the sequence of projector-valued measures associated to the $A_n$'s and let $E$ be the projector-valued measure associated to $A$.
Do we have, in some sense, a convergence $E_n \rightarrow E$?
For example, do we have, for all $x \in \mathbb{R}$, $E(\{x\}) = \bigcap_N \bigcup_{n} \bigcap_{k \geq n} E_k\left(\left[x-\frac{1}{N},x+\frac{1}{N}\right]\right)$?