Kind of convergence theorem for nets

Question

It is well known that the Lebesgue Dominated Convergence Theorem does not hold for nets (there are problems concerning countability). However, in the following particular case, is there any possible hypothesis, which I could add to the integrand function (actually on $f$) to switch the order of the limit and integral? $$ \lim_{\alpha \in A} \int_He^{-if(P_{\alpha}x)}d\mu=\int_H\lim_{\alpha \in A} e^{-if(P_{\alpha}x)}d\mu $$ Here $H$ is a Hilbert space ($\dim(H)=\infty$), $\mu$ some complex measure and $P_{\alpha}$ are projections converging strongly to the identity.