  We study the anomalous Hall effect in monolayers of transition-metal
dichalcogenides 2H-$MX_2$ ($M$=Mo, W, and $X$=S, Se, Te) under a proximity
effect of ferromagnetic substrate. If a proximity-induced exchange field is
introduced, the spin-polarized energy bands in $K$ and $K'$ valleys are shifted
in the opposite directions, and it causes the Hall effect by breaking
time-reversal symmetry. The induced Hall effect is the most prominent in the
valence band which has a large intrinsic spin splitting. Moreover, we find that
tilting the magnetization from the perpendicular direction gives rise to a
sensitive change in the Hall conductivity only in the electron side, and it is
attributed to the mixing of the Berry phase by the in-plane field in the nearly
degenerate conduction bands.
