  In this paper, the notion of measure complexity is introduced for a
topological dynamical system and it is shown that Sarnak's M\"{o}bius
disjointness conjecture holds for any system for which every invariant Borel
probability measure has sub-polynomial measure complexity.
  Moreover, it is proved that the following classes of topological dynamical
systems $(X,T)$ meet this condition and hence satisfy Sarnak's conjecture: (1)
Each invariant Borel probability measure of $T$ has discrete spectrum. (2) $T$
is a homotopically trivial $C^\infty$ skew product system on $\mathbb{T}^2$
over an irrational rotation of the circle. Combining this with the previous
results it implies that the M\"{o}bius disjointness conjecture holds for any
$C^\infty$ skew product system on $\mathbb{T}^2$. (3) $T$ is a continuous skew
product map of the form $(ag,y+h(g))$ on $G\times \mathbb{T}^1$ over a minimal
rotation of the compact metric abelian group $G$ and $T$ preserves a measurable
section. (4) $T$ is a tame system.

Dynamical Systems;Number Theory