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A system being measured (or interacting with its environment in any other way) is actually a subsystem, and a subsystem is properly described by a reduced density matrix. The density matrix of a pure state evolves according to the Liouville-von Neumann equation, which is equivalent to the unitary evolution of the wavefunction by the time-dependent Schrödinger equation. Sean Carroll's July Quick Study, "Addressing the quantum measurement problem" (page 62), brings up the following question: Does the wavefunction still obey the Schrödinger equation when a measurement is made?. [Extracted from the article]