  Gleason's theorem [A. Gleason, J. Math. Mech., \textbf{6}, 885 (1957)] is an
important result in the foundations of quantum mechanics, where it justifies
the Born rule as a mathematical consequence of the quantum formalism. Formally,
it presents a key insight into the projective geometry of Hilbert spaces,
showing that finitely additive measures on the projection lattice $\PH$ extend
to positive linear functionals on the algebra of bounded operators $\BH$. Over
many years, and by the effort of various authors, the theorem has been
broadened in its scope from type I to arbitrary von Neumann algebras (without
type $\text{I}_2$ factors). Here, we prove a generalisation of Gleason's
theorem to composite systems. To this end, we strengthen the original result in
two ways: first, we extend its scope to dilations in the sense of Naimark [M.
A. Naimark, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., \textbf{41}, 359 (1943)]
and Stinespring [W. F. Stinespring, Proc. Am. Math. Soc., \textbf{6}, 211
(1955)] and second, we require consistency with respect to dynamical
correspondences on the respective (local) algebras in the composition [E. M.
Alfsen and F. W. Shultz, Commun. Math. Phys., \textbf{194}, 87 (1998)]. We show
that neither of these conditions changes the result in the single system case,
yet both are necessary to obtain a generalisation to bipartite systems.
