Abstract: Given a topological dynamical system $(X,T)$ and a $T$-invariant measure $\mu$, let $\mathcal{B}$ denote the Borel $\sigma$-algebra on $X$. This paper proves that $(X,\mathcal{B},\mu,T)$ is rigid if and only if $(X,T)$ is $\mu$-$A$-equicontinuous in the mean for some subsequence $A$ of $\mathbb{N}$, and a function $f\in L^2(\mu)$ is rigid if and only if $f$ is $\mu$-$A$-equicontinuous in the mean for some subsequence $A$ of $\mathbb{N}$. In particular, this gives a positive answer to Question 4.11 in [1].}

Key words: Measure-theoretic equicontinuity, rigidity, mean metric