Integration with Spectral Measure

Question

Now, suppose I have a compactly supported spectral measure $E$ defined on the Borel subsets of $\mathbb{C}$. My question is this, suppose we have a sequence of functions $\{f_n\}$ non necessarily continuous, converging uniformly to $f$ continuous, is it true that $||\int f_{n}dE-\int fdE||\to 0$? I know the result holds in strong operator topology.

Answer 1

Yes. Write $E_h(\Delta)=\langle E(\Delta)h,h\rangle$. Then $$ \left\langle \left|\left(\int_{\mathbb C} (f_n(t)-f(t))\,dE(t)\right)h,h \right\rangle\right| =\left|\int_{\mathbb C}(f_n(t)-f(t))\,dE_h \right| \leq\int_{\mathbb C}|f_n(t)-f(t)|\,dE_h\leq\|f_n-f\|_\infty. $$ As you can do this for any $h$, $$ \left\|\int_{\mathbb C} (f_n(t)-f(t))\,dE(t)\right\|\leq\|f_n-f\|_\infty. $$