  After a brief introduction to the spectral presheaf, which serves as an
analogue of state space in the topos approach to quantum theory, we show that
every state of the von Neumann algebra of physical quantities of a quantum
system determines a certain measure on the spectral presheaf of the system. The
so-called clopen subobjects of the spectral presheaf play the role of
measurable sets. Measures on the spectral presheaf can be characterised
abstractly, and the main result is that every abstract measure induces a unique
state of the von Neumann algebra. Finally, we show how quantum-theoretical
expectation values can be calculated from measures associated to quantum
states.
