  The concept of $\mu-$equicontinuity was introduced by Gilman to classify
cellular automata. We show that under some conditions the sequence of Cesaro
averages of a measure $\mu,$ converge under the actions of a $\mu
-$equicontinuous CA. We address questions raised by Blanchard-Tisseur on
whether the limit measure is either shift-ergodic, a uniform Bernoulli measure
or ergodic with respect to the CA. Many of our results hold for CA on
multidimensional subshifts.
