Let $\mathcal{H}$ be a Hilbert space, $T$ a bounded self-adjoint operator, and $F:\left[a,b\right]\to\mathcal{B}\left(\mathcal{H}\right)$ such that for any $t\in\left[a,b\right]$, $F\left(t\right)$ is bounded and self adjoint. Let also $Q$ be a projection-valued measure. Suppose that $T$ commutes with $Q$, and that for any $t_1,t_2\in\left[a,b\right]$, $t\in\left[t_1,t_2\right]$ and $\psi\in\mathcal{H}$, $\|F\left(t\right)Q\left[t_1,t_2\right]\psi-TQ\left[t_1,t_2\right]\psi\|<\left|t_1-t_2\right|\|Q\left[t_1,t_2\right]\psi\|$. This property implies that the Riemann-Stieltjes integral $\int_{a}^{b}F\left(t\right)dQ\left(t\right)$ converges to $T$, and furthermore - that for any polynomial $p$, we have $\int_{a}^{b}p(F(t))dQ\left(t\right)=p(T)$.
I am trying to prove that this implies that if $A$ is a set which is of measure zero with respect to the spectral measure of $F\left(t\right)$ for any $t\in\left[a,b\right]$, then it is of measure zero with respect to the spectral measure of $T$. I don't know if it is true, but it does work in all of the simple examples ($\mathcal{H}$ is finite-dimensional, $Q$ has finite support). When $Q$ has finite support, for example, this implies that $H$ is a direct sum of the operators $F(t_1),\ldots,F(t_n)$ where $\left\{t_1,\ldots,t_n\right\}$ is the support of $Q$.
Any ideas on how to approach to this problem, or references to works which seem related?
