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To elucidate ideal measurements, one must explain how individual eventsemerge from quantum theory which deals with statistical ensembles, and howdifferent may end up with different final states. This so-called "measurementproblem" is tackled with two guidelines. On the one hand, the dynamics of themacroscopic apparatus A coupled to the tested system S is describedmathematically within a standard quantum formalism, where "q-probablities"remain devoid of interpretation. On the other hand, interpretative principlesare introduced to account for the expected features of ideal measurements. Mostof the five principles, which relate the quantum formalism to physical reality,are straightforward and refer to macroscopic variables. The process can beidentified with a relaxation of S+A to thermodynamic equilibrium, not only fora large ensemble $\scriptE$ of runs but even for its sub-ensembles. Thedifferent mechanisms of quantum statistical dynamics that ensure the relaxationare exhibited. The additional information provided by the sub-ensembles removeSchr\"odinger's quantum ambiguity of the final density operator for $\scriptE$which hinders its direct interpretation, and bring out a commutative behaviourof the pointer observable at the final time. The latter property supports theintroduction of a last principle, needed to switch from the statisticalensembles and sub-ensembles described by quantum theory to individualexperimental events. It amounts to identify some formal "q-probabilities" withordinary frequencies, but only those which refer to the final pointerindications. The desired properties of ideal measurements, in particular theuniqueness of the result for each individual run and von Neumann's reduction,are thereby recovered with economic interpretations. The status of Born's ruleinvolving both A and S is re-evaluated, and contextuality is made obvious.

A sub-ensemble theory of ideal quantum measurement processes