We consider measure theoretic equicontinuity and sensitivity via Furstenberg family. We introduce the notion of \(\mathcal {F}\) - \(\mu\) -equicontinuity which is the refined version of \(\mu \) -equicontinuity using Furstenberg family \(\mathcal {F}\) and prove that when \(\mathcal {F}\) is a filter, a given dynamical system \((X,T)\) is \(\mathcal {F}\) - \(\mu\) -equicontinuous if and only if it is \(\mathcal {F}\) - \(\mu\) - \(f\) -equicontinuous with respect to every continuous function \(f \colon {X \to \mathbb {C}} \) . In addition, under certain conditinos, we prove that an ergodic measure theoretic dynamical system is either \(k\mathcal {F}\) - \(\mu \) -sensitive or \(\mathcal {F}\) - \(\mu\) -equicontinuous.
