  Quantum Measure Theory (QMT) is a generalization of quantum theory where
physical predictions are computed from a matrix known as \emph{decoherence
functional} (DF). Previous works have noted that, in its original formulation,
QMT exhibits a problem with composability, since the composition of two
decoherence functionals is, in general, not a valid decoherence functional.
This does not occur when the DFs in question happen to be positive semidefinite
(a condition known as strong positivity). In this paper, we study the concept
of composability of DFs and its consequences for QMT. Firstly, we show that the
problem of composability is much deeper than originally envisaged, since, for
any $n$, there exists a DF that can co-exist with $n-1$ copies of itself, but
not with $n$. Secondly, we prove that the set of strongly positive DFs cannot
be enlarged while remaining closed under composition. Furthermore, any closed
set of DFs containing all quantum DFs can only contain strongly positive DFs.
