Three facts related to the collapse of quantum mechanical wavefunctions following measurements or observations are presented and explained. They are (1) in all stochastic theories, both classical and quantum mechanical, probability densities collapse following observations; (2) the laws of probability determine the probability density just after an observation in terms of that just before the observation and the result of the observation; (3) the wavefunction just after an observation can be found from that just before the observation by using the result of the observation, the laws of probability, and the assumption that immediate repetition of an observation yields the same result and does not change the wavefunction except for an arbitrary phase factor.
We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have 0–1 stochastic kernels. We next consider finite observables and instruments and show that in these cases, stochastic kernels are replaced by stochastic matrices. We also show that coarse-graining is the same as post-processing in this finite case. We then consider sequential products of observables and discuss the sequential product of a post-processed observable with another observable. We briefly discuss SIC observables and the example of qubit observables.
It is known that non-commuting observables in quantum mechanics do not have joint probability. This statement refers to the precise (additive) probability model. I show that the joint distribution of any non-commuting pair of variables can be quantified via upper and lower probabilities, i.e. the joint probability is described by an interval instead of a number (imprecise probability). I propose transparent axioms from which the upper and lower probability operators follow. The imprecise probability depend on the non-commuting observables, is linear over the state (density matrix) and reverts to the usual expression for commuting observables.