Measure theoretic equicontinuity and sensitivity via Furstenberg family

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.