  In this paper, we extend the classical Weyl's lemma to $RCD(K,N)$ metric
measure spaces. As its applications, we show the local regularity of solutions
for Poisson equations and a Liouville-type result for $L^1$ very weak harmonic
functions on $RCD(K,N)$ spaces. Meanwhile, a byproduct is that we obtain a
gradient estimate for solutions to a class of elliptic equations with
dis-continuous coefficients.
