  Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is
reductive. A flow $\phi $ of automorphisms of $Q$ endowed with an ergodic
probability measure on the compact base space $X$ induces two decompositions of
the flag bundles associated to $Q$. A continuous one given by the finest Morse
decomposition and a measurable one furnished by the Multiplicative Ergodic
Theorem. The second is contained in the first. In this paper we find necessary
and sufficient conditions so that they coincide. The equality between the two
decompositions implies continuity of the Lyapunov spectra under pertubations
leaving unchanged the flow on the base space.
