In 2014, together with D. Aerts, we showed - within the extended Bloch representation of quantum mechanics - that the Born rule can be understood as resulting from a universal average over all the possible ways in which the measurement-interactions can be selected, a result that was obtained by averaging over discretized (cellular) probability densities and then taking the continuum limit. The construction has since been questioned, in the context of Bertrand's paradox, by N.~Shackel, who observed that the result depends on the specific discretization adopted, that the sequence of averages is constant, and that the sense in which the average is taken over "all" probability densities remains unclear. In this article, we take these criticisms as a stimulus to re-examine the notion of universal measurement. We show that the key to the whole matter is to recognize that a universal measurement is a random way of selecting a measurement which does not privilege any portion of the abstract structure describing the available interactions, a property we call indifference. We prove that every indifferent universal measurement, however it is constructed, produces exactly the Born probabilities, so that the universal average is unique, and the cellular construction of 2014 is one among infinitely many ways of obtaining it. For cellular measurements, the result already holds exactly at every finite resolution, with no limit to be taken, which explains the constancy of the sequence of averages and dispenses with the combinatorial identities used in the original proof. We then show that non-trivial indifferent universal measurements exist, and that they must involve deterministic measurements: the random mixtures of deterministic measurements known in probability theory as Dirichlet processes are indifferent universal measurements whose support contains every possible way of selecting an interaction, whereas an indifferent universal measurement free of deterministic components necessarily selects the uniform density, so that for it the Born rule holds not only on average but with certainty. The deviations from the quantum statistics that deterministic components can produce in sequences of measurements are governed by a single parameter, which is in principle accessible to experiment. The only geometric input of the whole construction is the Euclidean length (or volume) of the measurement membrane, together with the orthogonal landing of the point-particle representing the state, which is where the content of Gleason's theorem lies. Finally, we discuss the physical meaning of these results, indicate how they generalize to measurements with an arbitrary number of outcomes, and comment on their bearing on Bertrand’s paradox.

[In preparation]