Most introductory text books on quantum mechanics have some variant of the following postulates for quantum mechanics: 1) The state of a system $S$ can be represented by a vector $\ket{\psi}$ in a Hilbert Space $H$. If $S$ can be decomposed into two smaller systems $S_A$ and $S_B$, which can be represented by Hilbert spaces $H_A$ and $H_B$ respectively, then $H = H_A \otimes H_B$ 2) $\ket{\psi}$ evolves according to a unitary operator determined by the <PERSON> equation except when measurements occurs. $\ket{\psi(t)} = U(t) \ket{\psi(0)}$. 3) When a measurement is performed, the state of the system collapses into a new state which corresponds to a definite result of the measurement. (Projection Postulate) 4) The probability of collapse to a given state is proportional to the magnitude of the overlap squared between the initial and final states (Born Rule).
Postulates 3&4, collectively referred to as the measurement postulates, are the source of most discomfort and confusion in quantum mechanics, and much controversy.
In the past two decades people have found several ways to derive the measurement postulates from axioms 1&2, often slightly modifying postulate #1, in an interpretation independent way. I personally love <PERSON>'s approach using envariance (see paper here, which is reasonably accessible). Here is another, more recent, derivation of the measurement postulates.
Many physicists (see <PERSON>'s paper "Against Measurement" for example) consider the traditional postules to be incomplete, because it doesn't include a prescription for what configurations of wavefunctions result in something that "counts" as a measurement. 976
For this reason, different modifications to the above postulates have been made over the years, and each of which allows different derivations of <PERSON> rule. These modifications are often called "interpretations" even though they are physically different theories which make theoretically testable different predictions (and sometimes actually testable, for example we've all but ruled out local hidden variables by testing Bell's Inequality). These include:
* Objective collapse theories Of these, I believe GRW is the most well developed, and there has been some interesting experimental work testing the theory in the last decade (see for example this)
* "No Collapse" theories. In this approach, only postulates 1 & 2 are assumed, and postulates 3&4 are derived from them under the additional assumption that the entire universe can be thought of as a quantum system. This is also called "many worlds" because, when deriving 3&4, you find that multiple classical realities are present in the wave function of the universe. Early efforts to derive <PERSON>'s rule within the no collapse approach were circular, but I don't believe this is the case for modern derivations. Derivation of <PERSON>'s rule in many worlds typically use <PERSON>'s theorem to "count" the number of different worlds where each outcome occurs.
* So many others (Consistent Histories, etc) 800