Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ the spectral measures of $A_n$ and $A$ respectively, what can be said about the strong limits of $\{E_n(\Omega)\}$, for $\Omega$ a Borel set in $\mathbb R$?

Supposing $\sigma_p(A) = \emptyset$, for every $a,b \in \mathbb R$ it holds $$ E_n((a,b)) \to E((a,b)) \mbox{ strongly} $$

What can be said in the case of a generic Borel set?


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