We prove that every closed
exhaustive vector-valued modular measure on a lattice ordered
effect algebra $ L $ can be decomposed into the sum of a Lyapunov
exhaustive modular measure (i.e. its restriction to every interval
of $ L $ has convex range) and an "anti-Lyapunov" exhaustive
modular measure.
\par This result extends a Kluvanek-Knowles decomposition theorem
for measures on Boolean algebras.