  We introduce a 'uniform tension-reduction' (UTR) model, which allows to
represent the probabilities associated with an arbitrary measurement situation
and use it to explain the emergence of quantum probabilities (the Born rule) as
'uniform' fluctuations on this measurement situation. The model exploits the
geometry of simplexes to represent the states, in a way that the measurement
probabilities can be derived as the 'Lebesgue measure' of suitably defined
convex subregions of the simplexes. We consider a very simple and evocative
physical realization of the abstract model, using a material point particle
which is acted upon by elastic membranes, which by breaking and collapsing
produce the different possible outcomes. This easy to visualize mechanical
realization allows one to gain considerable insight into the possible hidden
structure of an arbitrary measurement process. We also show that the UTR-model
can be further generalized into a 'general tension-reduction' (GTR) model,
describing conditions of lack of knowledge generated by 'non-uniform'
fluctuations. In this ampler framework, particularly suitable to describe
experiments in cognitive science, we define and motivate a notion of 'universal
measurement', describing the most general possible condition of lack of
knowledge in a measurement, emphasizing that the uniform fluctuations
characterizing quantum measurements can also be understood as an average over
all possible forms of non-uniform fluctuations which can be actualized in a
measurement context. This means that the Born rule of quantum mechanics can be
understood as a first order approximation of a more general non-uniform theory,
thus explaining part of the great success of quantum probability in the
description of different domains of reality. This is the first part of a
two-part article.
